21.4.5.2. Multi-Objective Optimization

In mathematical form, multi-objective can be defined as

\(\underset{\mathbf{x}}{\mathop{\min }}\,P\left( \mathbf{f}\left( \mathbf{x} \right) \right)\)

where, \(\mathbf{f}\left( \mathbf{x} \right)\) is a vector type objectives and the functional \(P\left( \cdot \right)\) is called as a preference function. The preference function is an equivalent functional that transform the vector type objective into a scalar type objective. In order to represent our preference function, let’s consider following two objectives.

\(\underset{\mathbf{x}}{\mathop{\min }}\,{{f}_{1}}\left( \mathbf{x} \right)\) and \(\underset{\mathbf{x}}{\mathop{\max }}\,{{f}_{2}}\left( \mathbf{x} \right)\)

There are many preference functions in multi-objective optimization strategy. Among them, RecurDyn/AutoDesign uses following two types.

\(P\left( \mathbf{x} \right)={{w}_{1}}\left( \frac{{{f}_{1}}\left( \mathbf{x} \right)}{\eta f_{1}^{G}}-1 \right)+{{w}_{2}}\left( 1-\frac{f{{\left( \mathbf{x} \right)}_{2}}}{\nu f_{2}^{G}} \right)\)

\(P\left( \mathbf{x} \right)=\max \left\{ {{w}_{1}}\left( \frac{{{f}_{1}}\left( \mathbf{x} \right)}{\eta f_{1}^{G}}-1 \right),{{w}_{2}}\left( 1-\frac{f{{\left( \mathbf{x} \right)}_{2}}}{\nu f_{2}^{G}} \right) \right\}\)

where, the values of \({{w}_{i}}\) are the user defined weighting coefficients and the relaxation factors \(\eta\) and \(\nu\) are automatically determined. Also, the ideal solution \(f_{i}^{G}\) is internally determined.