## 5 Yield Optimization
### Designs with Failure Allowance  
In this section, yield optimization for determining a nominal DGR design point is exemplified by assuming that the UFC spacing (\(x_{1}\)) and room spacing (\(x_{2}\)) of the repository are random variables with arbitrary tolerances and distributions. Due to a single variable representing the UFC spacing between all containers and the COMSOL model portraying an infinite repository, the interpretation of the results should consider that the realized values of the random variables represent that of the entire repository. Because the bounds considered on UFC and room spacings would result in a large feasible region (e.g., see Figure 6), yield optimization would always give 100% yield designs for any reasonably sized tolerance box on the two random variables, regardless of their distributions. However, should some failure probability be allowed such that a portion of the design's tolerance box can extend across the feasible boundary, the distributions of the random variables would then be significant. Thus, for demonstration, we studied failure probabilities of 1% and 5% for the specific case of an assorted UFC age inventory arranged using a Kumaraswamy PDF-based shape function with shape parameter values of \(a=0.25\) and \(b=1.5\). Note that earlier in Section 4.3.2 and Figure 8, it was shown that the largest UFC spacings are required for shape parameter values of \(a\to 0.25\) because this would mean like-aged UFCs being densely placed. Therefore, the subsequent calculations can be considered as designing for unfavorable or high-risk cases. To obtain the feasible region required to start yield optimization, \(R_{A}\) in Equation (7), surrogate-based optimization was first performed for \(T_{\max}=T_{\max}(x_{1},x_{2})\) with \(a=0.25\). The objective function in Equation (12) was modified to incorporate failure probability. Additionally, a penalty term that discouraged large UFC spacings and room spacings was included for reducing the DGR construction cost; otherwise, optimization would place the tolerance box in the center of the large feasible region, thus leading to no failures but a huge cost to construct the DGR. The yield optimization problem became:  
\[ \begin{array}{ll}\max_{\mathbf{x}^{\prime},\mathbf{x}^{\prime},\mathbf{x}^{\prime\prime}}& \text{Yield}(\mathbf{x}^{\prime},\mathbf{x}^{\prime},\mathbf{x}^{\prime\prime})-\frac{1}{100} \left(\frac{\theta\left(X_{0,1}-x_{1}^{\prime}\right)}{x_{1}^{\prime\prime}-x_ {1}^{\prime}}+\frac{\left(1-\theta\right)\left(X_{0,2}-x_{2}^{\prime}\right)} {x_{2}^{\prime\prime}-x_{2}^{\prime\prime}}\right)\\ \text{subject to}&\text{Yield}\left(\mathbf{x}^{\prime},\mathbf{x}^{\prime},\mathbf{x}^{ \prime\prime}\right)\leq 1-F^{T}\\ &\mathbf{A}^{+}\mathbf{x}^{\prime\prime}-\mathbf{A}^{-}\mathbf{x}^{\prime}\leq\mathbf{c}\\ &\mathbf{x}^{\prime\prime}\leq\mathbf{x}^{\prime}+\mathbf{t}\\ &\mathbf{x}^{\prime}\leq\mathbf{x}^{\prime}\leq\mathbf{x}^{\prime\prime}\end{array}  \tag{14}\]  
where \(\theta\) is an arbitrary weighting parameter, \(\mathbf{X}_{0}\) is the nominal design point, and \(F^{T}\) is a target failure probability. The constant \(1/100\) scales the penalty term down so that maximizing yield is of higher priority. The arbitrary weighting parameter determines whether tighter UFC spacing or room spacing is more important. Since larger UFC spacing increases the DGR footprint and the amount of host rock excavated for more placement rooms while larger room spacing only increases the DGR footprint, preference was given to smaller UFC spacings by assigning the parameter a value of 0.8 so that designs that reduce the overall cost of the repository would be identified (compare the excavation cost in [22] and land cost in [23]). The modification of including the target failure probability could lead to inaccurate results because the predicted yield only approximates the actual portion of the tolerance box within/beyond the feasible region. However, for small failure probabilities, the approximation should be reasonable.