**5.1. Pareto-optimal solutions**

224 Simulated Annealing – Single and Multiple Objective Problems

**Figure 4.** Mass densities and displacement functions for FGM dental implant: (a) cortical density function (*f1*), (b) cancellous density function (*f2*), (c) displacement function (*f3*) (The horizontal axis is *m*).

The multi-objective optimization has become an important research topic for scientists and researchers. This is mainly due to the multi-objective nature of real life problems. It is difficult to compare results of multi-objective methods to single objective techniques, as there is not a unique optimum in multi-objective optimization as in single objective optimization. Therefore, the best solution in multi-objective terms may need to be decided

**5. Multi-objective optimization** 

by the decision maker.

The concept of the Pareto-optimal solutions was formulated by Vilfredo Pareto in the 19th century (Rouge, 1896). Real life problems require simultaneous optimization of several incommensurable and often conflicting objectives. Usually, there is no single optimal solution. However, there may be a set of alternative solutions. These solutions are optimal in the wider sense that no other solutions in the search space are superior to each other when all the objectives are considered. They are known as Pareto-optimal solutions. When the objectives associated with any pair of non-dominated solutions are compared, it is found that each solution is superior with respect to at least one objective. The set of non-dominated solutions to a multi-objective optimization problem is known as the Pareto-optimal set (Zitzler & Thiele, 1998).
