## 3 Linking Early Robust Design and Optimal Parameter and Tolerance Design
### Effects of Robust Concept Design on Parameter and Tolerance Optimization
The optimization problems can generally be defined as follows [46; 71]:  
\[\text{optimal parameter design:}\] least-cost tolerance design: \[\text{Minimize} \Delta\mu_{k}, C_{\text{sum}}(\mathbf{t})=\sum_{i=1}^{I}C_{i}(t_{i}), \tag{1}\] \[R_{k}(\mathbf{X})\ \forall\ k=1,\ldots,K,\] subject to \[g_{i}(\mathbf{X})\leq 0,\ \forall\ i=1.\ldots,I, \hat{z}_{k}\leq z_{\text{max},k}\ \forall\ k=1,\ldots,K,\] (2) \[h_{j}(\mathbf{X})=0,\ \forall\ j=1,\ldots,J. \hat{z}_{\text{tot}}\leq z_{\text{max}}, \tag{3}\]  
with \(\Delta\mu_{k}\): mean shift between target value and mean of scattering of KC \(k\); \(R_{k}\): robustness metric for KC \(k\); \(C_{\text{sum}}(\mathbf{t})\): total costs for a specific set of tolerances; \(g_{i}\): inequality constraints; \(h_{j}\): equality constraints; \(\hat{z}_{k}\): single scrap rate for KC \(k\); \(\hat{z}_{\text{tot}}\): total product scrap rate. As a consequence, the optimization problem and its solution mainly depend on the number (\(K\)) and interrelation of the KCs as well as their mathematical description. While in parameter design, the KCs are influencing the objective(s), the aim of optimal tolerance design leads to a shift of their consideration in the constraints, see Equation (1). Thus, the two axioms with the associated uncoupling and information reduction in the concept design affect the robust parameter optimization as well as the tolerance-cost optimization. Figure 4 gives a comprehensive summary of these effects. The uncoupling of the KCs in the concept design (Axiom 1) results, as already demonstrated in the center in Figure 3, in completely independent KCs, each of which can be described separately. Thus, the transfer functions usually used to describe the mathematical relationship between varying parameters and the respective KC are also uncoupled so that all the parameters of the product always influence only one KC. This separate formulation of the underlying functional relation representing the product behavior simplifies the overall optimization problem. Thus, the multi-objective robust parameter optimization, which is necessary to consider the \(K\) interrelated KCs, see Equation (1), can be split into several uncoupled optimization problems for each individual KC. As a consequence, trade-offs, which usually require a weighting and prioritization of the analyzed KCs or a manual selection of the optimal result, e.g., from a Pareto front, are omitted. Thus, unambiguous optimal solutions result for each KC, since no compromises between potentially contradicting KCs are necessary. In addition, this leads to a reduced number of parameters that need to be considered in each single optimization. In the example in Figure 3, the number is thus lowered from five to three parameters each, leading to a splitting of the five-dimensional optimization space into two three-dimensional spaces. In the case of tolerance-cost optimization, the simplified optimization problem affects the constraint definition, see Equations (2) and (3) and Table 2. The uncoupling of the KCs leads to an elimination of the dependencies between the individual constraints, thus enabling a separate optimization on the basis of the single transfer functions. This, consequently, leads to a less restricted search space since the individual tolerances can be chosen with respect to just one single KC. The resulting additional optimization potential allows an independent selection of nominal design parameters and tolerances. Regardless of any sampling-based or optimization algorithm-related anomalies, this finally contributes to better optimization results in terms of robustness and costs. According to the hypothesis stated above, reducing the number of elements involved in a KC (Axiom 2) in the conceptual design stage indicates a smaller number of varying parameters in the transfer function describing the KC in the subsequent optimization process. The lower number of parameters contributing to the respective KCs leads to a reduced variation of the KCs and thus to a less noisy optimization problem.