## 5 Application of the Approach
### Prioritization of the Risk Treatment paths  
For the measure selection, the AHP periodization matrix and utility functions for the three criteria of cost, risk reduction, and energy flexibility are needed. An interactive MATLAB(r) Live Script is created as the user interface. For optimization, MATLAB(r) offers an Optimization Toolbox(tm) that can solve MILPs efficiently. The approach is implemented in MATLAB(r) version 9.6.0.1472908 (R2019a). All input data are stored in Microsoft(r) Excel(r) and can therefore be easily modified. Firstly, all ratings for the paths created in advance using the approach in [PERSON] et al. [6] are normalized, resulting in trilemma criteria for every path. In Figure 6, the trilemma for  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  
three exemplary paths is depicted. To compare the three paths, their criteria are shown in the diagram below on the right. With the involvement of the experts from the production system, the approach to weighting the trilemma criteria was then carried out in order to be able to categorize the paths according to their suitability. For the decision matrix in Table 4, the principal eigenvector according to the AHP [23] determines the weighting of 0.0679 for cost, 0.7703 for energy flexibility, and 0.1618 for risk reduction, according to the assessment of the experts. This is due to a high need for the remaining energy flexibility in the system for reactive measures, since the orders are particularly tightly timed in the period under consideration, and failure to meet deadlines is associated with high penalties. Risk reduction is also given a higher weighting than cost; additional costs for measures are therefore accepted in favor of risk reduction. The objective weighting, applying Shannon entropy, results in a weighting of 0.0245 for cost, 0.0710 for energy flexibility, and 0.9044 for risk reduction, assuming all values are normalized according to Equation (5) or Equation (6). The proportionally high weighting of risk reduction in comparison to the two remaining criteria highly influences the final weighting obtained by multiplicative aggregation. The large deviation in values is not uncommon, as subjective and objective weighting are generated independently. Thus, the final weighting results in 0.0082 for cost, 0.2700 for energy flexibility, and 0.7218 for risk reduction. For this use case, the utility functions based on the underlying data structure were selected, so that differences in ratings are amplified. This is achieved by choosing a convex utility function when most data points lie within the upper third of the scale. When most date points lie within the lower third of the scale, a concave utility function is applied. For \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Cost** & **Energy Flexibility** & **Risk Reduction** & **Final Weighting** \\ \hline
**Cost** & 1 & 1/9 & 1/3 & 0.0679 \\
**Energy** & 9 & 1 & 6 & 0.7703 \\
**flexibility** & 3 & 1/6 & 1 & 0.1618 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Decision matrix for the AHP. Figure 6: Trilemma of three risk treatment paths including an overview. the remaining third, a linear function was utilized. Thus, for cost and energy flexibility, a convex utility function was chosen and for risk reduction a linear utility function. \[u_{ij}\Big{(}r_{ij}^{*}\Big{)}=\frac{exp\Big{(}\big{(}r_{ij}^{*}\big{)}^{2}\Big{)} -1}{exp(1)-1} \tag{22}\]  
Applying the respective utility function to the decision matrix results in marginal utility scores for every alternative and every criterion. The final utility scores are calculated by weighted sums, and the paths, i.e., the alternatives, are ranked in descending order. Table 5 depicts the top three paths with their marginal and final utility scores under the assumption of the previously generated final weighting.