## 3 Linking Early Robust Design and Optimal Parameter and Tolerance Design
### Comparison of Characteristic Aspects
Despite the fact that the basic idea and the characteristics of the optimization domains are very similar, they differ in the definition of the objectives and constraints (also see Figure 1) and thus also in the mathematical formulation of the optimization problem. While KC variation is considered as an objective in robust parameter optimization, predefined limits of KCs usually form the constraint of tolerance-cost optimization, see Table 2.  
In summary, it can be stated that the three domains differ significantly despite the consistency in the optimization approaches. This becomes particularly clear when comparing the _general aims_ of the respective approaches, namely robust concept design, robust nominal design parameters and cost-optimal tolerance values. As the comparison in Table 2 shows, the various aspects in the domains are therefore considered differently. However, it can be seen that the domains depend on each other and that the common aim of improved robustness of the product to variation is achieved at best by a sequential combination of the domains. This inevitably requires knowledge about the interactions between the corresponding approaches. \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Aspect** & **Early Robust Design** & **Robust Parameter Optimization** & **Tolerance-Cost Optimization** \\ \hline
**domain characteristics** & & & \\ extent number of perspectives & entire system & specific aspects & specific aspects \\ number of perspectives & high (design, parameter, variation) & medium (parameter, variation) & medium (variation) \\ dimension & primarily 2D & 2D/3D & 2D/3D \\ functional system description & qualitative & quant. transfer function & quant. transfer function (implicit/explicit) & (implicit/explicit), quant. tolerance-cost function, (quality loss function) \\ level of detail & low & medium & high \\ \hline
**control factors** & & & \\ engineer & design engineer & product simulation engineer, design engineer & tolerance engineer, design engineer \\ environment / department facilities & design, concept development & design engineer & tolerance management \\ specific method & n.a. (workflow) & computer-aided tools & computer-aided tools \\ \hline
**noise factors** & & & \\ uncertainty & epistemic uncertainty in designer decision & reduced ambiguity, aleatory uncertainty & primarily aleatory uncertainty \\ informal knowledge & epistemic design knowledge & n.a. & n.a. \\ \hline
**input** & & & \\ formal knowledge availability & low & high & high \\ information—data & requirements, initial concept and product structure & embodiment design & nominal design, manufacturing information \\ information—documents & sketch, graph & product model, (CAD/drawing) & product model, (CAD/drawing) \\ boundary conditions & requirements & design constraints, parameter ranges, noise & quality requirements, tolerance ranges, KC \\ \hline
**output** & & & \\ quantitative objective & n.a. & KC (mean shift and robustness) & tolerance-related costs \\ objective metrics & n.a. & robustness metrics & tangible and intangible costs cost-optimal tolerance values \\ general aim & robust concept design & robust nominal design parameters & tangible and intangible costs cost-optimal tolerance values \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the domains early robust design, robust parameter and tolerance-cost optimization.