## 2 Related Work
### Robust Design in Product Development
#### 2.1.2 Late Robust Design  
In contrast, late robust design, which is assigned to the embodiment and detail design stage, uses primarily quantitative approaches for parameter and tolerance design as well as robust design optimization. Thus, this traditional robust design mainly deals with Design of Experiments (DoE), statistics and optimization [5; 30; 31]. Therefore, transfer functions, which represent the relation between design parameters and product behavior, are determined by experiments or simulations and evaluated [32]. The product behavior is mapped by the quantifiable parameters respectively characteristics. In this context, they are often called Key Characteristics (KC) representing the relevant product requirements, e.g., via a distance between two adjacent surfaces [33]. The subsequent evaluation of the resulting KC enables the determination of the expected quality. Therefore, quality metrics, such as statistical moments, e.g., mean or variance, the scrap rate, the yield or process capabilities, are suitable [34]. In addition, Taguchi's quality loss functions enable the quantification of the impact of varying KCs on customers' perceived quality [35], thus linking to associated costs [34]. Further metrics, such as the Signal-to-Noise Ratio (SNR) [36] or sensitivity indices [37; 38], allow a quantification of the robustness [39], supporting engineers in the stringent optimization of parameter and tolerance values [40]. During parameter design, the nominal design parameter values, e.g., the length of a lever, are first modified so that the KC becomes insensitive to design parameter variations. Considering the inherent variance of design parameters, this leads to a mean shift in the KC resulting in improved quality without influencing product costs [25]. During the subsequent tolerance design, the admissible variation of design parameters is defined with respect to the predefined KC tolerance. However, the modification of design parameter tolerances is directly linked to manufacturing costs [41], which leads to a tolerance-cost conflict. Since established approaches for solving this conflict are not universally applicable [34], tolerance values are commonly adjusted iteratively in a time-consuming process, often failing to achieve the optimal values [42; 43; 44; 45]. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Sub’s Axiom** & **Robust Design Principle** & **Classification** \\ \hline \multirow{6}{*}{Axiom 1} & prevention of overconstrained systems by proper definition of constraints and degree of freedom [27] \\  & unambiguous design of interfaces, e.g., by sufficient clearance to avoid unintended \\  & contact [8] \\  & division of tasks [28] \\  & decoupling and uncoupling [29] \\ \hline \multirow{3}{*}{Axiom 2} & shortening of force-transmission paths [28] \\  & minimizing the number of design parameters [21] \\  & shielding from the cause of variation (e.g., heat) [21] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Assignment of robust design principles to [PERSON]’s axioms. In order to overcome these drawbacks, computer-aided, optimization-based approaches are increasingly used in parameter and tolerance design [34; 46]. Therefore, besides individual approaches, existing tools from research and industry can be utilized [47; 48]. This requires a transfer of the search for optimal nominal design parameter and tolerance values into a mathematical optimization problem [25; 34]. Although the basic formulation of the parameter and tolerance design problem is largely the same [46], they differ in the definition of objectives and constraints in combination with the used quality and robustness metrics, see Figure 1. In robust parameter optimization, often called robust (design) optimization, the robustness of the solution is either added to (a) the constraints or to (b) + (c) the objective(s) [25]. The parameter optimization tries to satisfy both the requirement for the mean value of the resulting scattering KC and the requirement for robustness or sensitivity of the KC around this mean value.