## 4 Description of the Approach
### Prioritization of the Risk Treatment Paths with a Hybrid MCDM Approach
#### 4.1.2 Subjective and Objective Criteria Weighting  
For the subjective expert weighting, the three trilemma criteria AHP is applied. The AHP [23] is widely used in the literature and practice due to its straightforward application and high reliability, even in uncertain decision situations [47]. It is based on the principle of pairwise comparison, usually done by experts, and it serves as the subjective weighting method in the approach. The pairwise comparison results in an \(n\times n\) decision matrix A with the elements \(a_{ij}\), each of which indicates the relative weighting of two criteria, \(i\) and \(j\). Values on the diagonal are equal to one. The remaining comparisons are filled with values from one to nine, and their inverse fractions for opposite dependencies [48]:  
\[A= \begin{bmatrix}1&\cdots&a_{1n}\\ \vdots&\ddots&\vdots\\ \frac{1}{a_{1n}}&\cdots&1\end{bmatrix} ,\;a_{ii}=1,\;a_{ji}=\frac{1}{a_{ij}}, \;a_{ij}\
eq 0 \tag{8}\]  
The following applies for the pairwise comparison: the higher the chosen value for criterion \(i\), the higher the preference of criterion \(i\) over criterion \(j\). For the comparison, a scale from one to nine is introduced, as it is effective in expressing preferences with sufficient precision and does not overwhelm the human decision-maker [49]. A verbal explanation of the nine levels is given in Table 2.  
The choice of the exact preference level within the range is the decision-maker's. As outlined in Section 4.1.1, the three relevant criteria to compare are cost, EF, and risk reduction. The regarded factors, which influence the pairwise comparison, are, e.g., a client's importance, sustainability goals, risk attitude, and time criticality. Some examples of decisions are listed below:  
* Risk reduction is more important than cost: the orders are particularly time-critical and for particularly important customers. Additional costs are therefore accepted for the reduction of risks. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Intensity of Importance on an** & **Definition** & **Explanation** \\ \hline
1 & Equal importance & Two activities contribute equally to the objective \\
3 & Moderate importance of one over another & Experience and judgment slightly favor one \\
5 & Essential or strong importance & activity over another \\
7 & Very strong importance & An activity is strongly favored and its dominance \\
9 & Extreme importance & The evidence favoring one activity over another \\  & Intermediate values between the two & of the highest possible order of affirmation \\
2, 4, 6, 8 & adjacent judgments & When compromise is needed \\  & If activity \(i\) has one of the above numbers assigned to it when compared with activity \(j\), then j \\ Reciprocals & has the reciprocal value when compared with \(i\). \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Descriptive explanation of the AHP pair-wise comparison scale [49]. * Energy flexibility is more important than risk reduction: the information available in the planning period is vague because rush orders could arrive. High remaining energy flexibility is required in the system. * Cost is more important than risk reduction: the forecasts for penalties for deviations are low, and load peaks are not expected in the period under consideration. There is no great need to incur costs for risk reduction measures. These comparisons can serve as a guide for decision-makers. Individual preferences may exist in a specific production system, and thus individual decisions must take into account the circumstances and requirements of the production system in question. The calculation of the criteria weights \(w_{AHP}\) for the decision matrix A is carried out by multiplying its largest eigenvalue \(\lambda_{max}\) by the respective eigenvector \(\overrightarrow{v_{i}}\):  
\[w_{AHP}=\lambda_{max}\overrightarrow{v_{i}},\ \ \text{with}\ \lambda_{max}= \underset{i}{\text{max}}\lambda_{i} \tag{9}\]  
Finally, the consistency of the expert judgments is checked with the consistency ratio (CR).