# Portfolio Implementation Risk Management Using Evolutionary Multiobjective Optimization
## 2 Materials and Methods
### Proposed Approach
#### 2.2.2 Fitness Assignment
It can be defined as the Lebesgue measure \(\Lambda\) of the union of hypercuboids in the objective space:  
\[\mathcal{S}(\mathcal{F})=\Lambda\left(\bigcup_{a\in\mathcal{F}}[f_{1}(a),r_{ 1}]\times\cdots\times[f_{m}(a),r_{m}]\right) \tag{8}\]  
before calculating hypervolume contributions of individuals in the corresponding front, the objective values, to be minimized, are normalized as:  
\[\bar{f}_{j}=\frac{f_{j}-f^{\min}}{f^{\max}-f^{\min}}\quad\forall j\in\{1, \ldots,m\}, \tag{9}\]  
where \(\bar{f}_{j}\) is the normalized objective value, \(f^{\min}\) and \(f^{\max}\) are the minimum and maximum objective values of the \(j\)-th objective in the corresponding front, respectively. This way, the objective values are mapped into the domain \([0,1]\), whereas the reference point is \(\overrightarrow{\mathbf{1}}\). The extreme points are emphasized by assigning a maximum possible value of \(\Delta_{\mathcal{S}}\). The fitness assignment procedure calculates ranks for individuals in each front based on the values of hypervolume contribution \(\Delta_{\mathcal{S}}\) and robustness \(R\) (lines 14-21). The use of rank values allows for coping with different magnitudes of measures incorporated into the fitness. With rank values, it is easy to aggregate the convergence, diversity and robustness measures, regardless of whether increase or decrease in their values leads to a better performance. In the case of \(\Delta_{\mathcal{S}}\), higher values are preferred. Thus, an individual with the largest value of \(\Delta_{\mathcal{S}}\) is assigned \(rank_{\Delta_{\mathcal{S}}}=1\). Next, the remaining individuals are considered, with \(rank_{\Delta_{\mathcal{S}}}=2\) being assigned to an individual having the largest value of \(\Delta_{\mathcal{S}}\) and so on. On the other hand, lower values of \(R\) correspond to more robust solutions. Thus, the ranking based on the robustness measure values is performed in ascending order. An individual with the smallest value of \(R\) is assigned \(rank_{R}=1\), an individual having the second smallest value of \(R\) being assigned \(rank_{R}=2\) and so on. Finally, the global fitness value \(F\) is computed for each population member. This is done by aggregating the three ranks as shown in line 23. Thus, the resultant fitness value reflects the quality of a given individual with respect to convergence, diversity and robustness. The extent to which the robustness influences the fitness of population members is determined by the dispersion parameter \(\epsilon\), which is a control parameter set by the user in the range \([0,1]\). Increasing \(\epsilon\) will weak the role of robustness, with the population diversity in the objective making more important. On the contrary, reducing \(\epsilon\) will put more emphasis on the solution robustness. As a result, selection is expected favor more robust individuals and to focus the search on regions with higher robustness of solutions. For \(\epsilon=1\), the selection reduces to the original procedure of SMS-EMOA [15], which can be viewed as a particular case of the proposed selection procedure. The dispersion parameter \(\epsilon\) is an important feature of the proposed approach. It enables the user to express his preferences about the robustness of solutions, thereby determining the focus of the search in advance. This way, a high-grained approximation of the preferred part of the efficient set can be obtained. This is a useful property, as, in practice, many Pareto optimal solutions are not relevant for the decision maker. Typically, the decision maker is only interested in a subset of optimal solutions that exhibit certain characteristics. Moreover, a single solution is eventually selected by incorporating some high-level preferences. By enabling the user to integrate his preferences before the search, the proposed approach aims to support decision-making when determining an optimal strategy for investment in a portfolio of assets.