## 2 Identify Complexity Characteristics
### Mathematical Approach to Describe and Calculate the Complexity  
The assignment of the different characteristic values to certain complexities can be conducted in different ways. On the one hand, it is possible to conduct an expert survey on the basis of which the individual characteristic values are assigned to certain complexity evaluations. This is particularly advisable if the individual characteristic values are not well described in the literature or only theoretically, if they are not physically measurable or if the product is so new that they are not physically present. It is also possible to evaluate the characteristic values based on individual criteria. For this purpose, evaluations with a scale of at least 1-5 (based on [24; 25]) are particularly suitable. These ratings are particularly advantageous when many different criteria describe a characteristic or variant. If explicit physically measurable or quantifiable values are available for a characteristic value, it makes sense to relate these directly to complexity.  
Further possibilities for evaluating different characteristic values with regard to the complexity they induce are as follows: performance of a utility value analysis, purely subjective assessment and sensitivity analyses.  
Mathematical models serve as a basis for identifying and defining the individual categories of complexity features and the associated complexity. For this purpose, the minimum and maximum value of the characteristic must be determined and the relationship between complexity and characteristic must be chosen. Figure 3 details the possible corre  
Figure 2: Detailing of the procedure for the development of a specific complexity indicator.  
lations between characteristic value and complexity and how such a relationship can be described mathematically.  
The first step is to identify the limits of the characteristic value and, thus, the maximum and minimum possible value the characteristic can take. Then, the correlation between complexity and characteristic can be described as a mathematical function.  
For linear relationships, only the minimum and maximum values with associated complexity values are required. In the case of exponential progressions, an average complexity must be specified in addition to these values, i.e., the value of the characteristic value at which the complexity is 0.5. For parabolic relations, values for the minima and maxima of the characteristic value have to be specified. In the case of staircase or jump functions, an explicit complexity value must be assigned to each characteristic value. After this step, all complexity characteristics are available with their values (in groups, if necessary) and the corresponding complexity values.  
Not every complexity characteristic has the same influence on product complexity. In order to determine this influence, a simple pairwise comparison of the characteristics can be used. Using this comparison, individual characteristics can be weighed against each other and, thus, be included in the overall calculation of the product complexity. This allows a more precise result and a more detailed consideration of the individual complexity characteristics in terms of their influence on overall product complexity. As a final step in the procedure, the individual complexity indicators (the individual complexity describes the specific partial complexity of the part of the product or system or the element under consideration) of the different levels of observation (describes the part of the product or system or the element under consideration, e.g., use case battery system: electrode, cell,  
Figure 3: Possible correlations of the classification of different characteristic values.  
module or pack level) are combined. Thus, the total complexity (\(K_{G}\)) can be calculated on the basis of partial complexities (\(K_{i}\)) and the individual weightings (\(y_{i}\)) and can be determined by using the following formula.  
\[K_{G}=\frac{\sum_{i=1}^{n}y_{i}\cdot K_{i}}{\sum_{i=1}^{n}y_{i}} \tag{1}\]