## 3 Linking Early Robust Design and Optimal Parameter and Tolerance Design
### Effects of Robust Concept Design on Parameter and Tolerance Optimization  
Consequently, this section analyzes the effect of early robust design decisions on robust parameter and tolerance-cost optimization. However, since the principles underlying the early robust design are mostly attributable to a desired uncoupling or reduced information content according to the _Independence or Information Axiom_, see Table 1, the following analysis focuses on the effect of uncoupling and information reduction in concept design on the subsequent optimization process. Figure 3 illustrates the relationship between the axioms and the resulting design by means of a simplified stacked blocks case study (bottom) with corresponding matrix and graph representation. Compared to the usual notation in Axiomatic Design, this representation is adapted to the robust concept design context by using KC instead of functional requirement and the assumption that information reduction is achieved by reducing the number of elements contributing to a KC. Although this assumption is not always valid in general [68], it is reasonable in the context of robust concept design [23], since further information is usually lacking at this stage, which leads to a higher probability of shorter tolerance chains or less information content in the case of fewer elements. The analysis of the effects for the optimization requires a detailed look on the respective mathematical optimization problems, already briefly discussed in Figure 1. In robust parameter design, the main goal is to improve the global system robustness by a conscious selection of the nominal design parameters \(X_{i}\) within predefined lower and upper boundaries \(X_{i}^{\text{lb}}\) and \(X_{i}^{\text{ub}}\). Focusing on case (c) in Figure 1, the mean shift \(\Delta\mu_{k}\) from the target value and the robustness metric \(R_{k}\), such as variance or the sensitivity, are forming the objectives covering the individual \(K\) KCs. Thereby, the exact mathematical formulation and solution of the optimization problem strongly depend on the chosen robustness metric. In the literature, different methods were presented to statistically evaluate the effects of aleatory uncertainties on the system robustness. Thus, the effects of variation of the currently chosen nominal design parameters, e.g., with respect to its general tolerances [69], on the KC can be measured in the optimization using these metrics. Additional inequality constraints \(g_{i}\) and equality constraints \(h_{j}\) are necessary to cover design restrictions and to avoid technically infeasible solutions. In the subsequent tolerance-cost optimization, it is the aim to identify the least-cost tolerances \(t_{i}\) within predefined tolerance ranges [34], see Figure 1e. Tolerance-cost optimization thus corresponds to the minimization of the total costs \(C_{\text{sum}}\) as the sum of all tolerance-related costs \(C_{i}(t_{i})\)[34]. Tolerance analysis is incorporated into the constraints to  
Figure 3: Influence of the _Independence Axiom_ (Axiom 1) and the _Information Axiom_ (Axiom 2) on concept design. assure product quality with respect to the predefined \(K\) KCs. In doing so, a single scrap rate or non-conformance rate for each KC as well as a total product scrap rate \(\hat{z}_{\text{tot}}\) are commonly used to ensure product functionality [70; 71].