## 4 Surrogate-Based Optimization
### Optimal Arrangements for an Inventory with Identical UFC Age  
For the case of an inventory with identical UFC age (\(x_{3}=30\)), the maximum temperature within the DGR was found to be a function of the spacing design variables, i.e., \(T_{\max}=T_{\max}(x_{1},x_{2})\). The surrogate-based optimization of Equation (2) when a cubic polynomial surrogate function was used resulted in the determination of the black boundary in Figure 6a, which approximated the combinations of UFC and room spacings that would result in maximum temperatures of 100 \({}^{\circ}\)C. Including the design variable bounds, the feasible region in Figure 6a is the region above the black boundary.  
Recalling Equation (1), the optimal design points are those closest to and above the black boundary. For a UFC inventory assumed to be identical in age, the discrete optimal combinations of UFC and room spacings are indicated in Figure 6b. The results reveal that if spacer blocks are not used in the DGR (i.e., \(x_{1}=1.0\)), the minimum room spacing required to keep the maximum temperature below 100 \({}^{\circ}\)C is 27 m. If spacer blocks are at their widest considered width of 2 m (i.e., \(x_{1}=2.0\)), the room spacing can be reduced to 12 m.  
Figure 6: (a) Surrogate-based optimization of Equation (2) for UFCs, all of 30 years of age, using a cubic polynomial surrogate function resulted in the black boundary representing the maximum temperature constraint. (b) Optimal design points as per Equation (1) are those closest to and above the black boundary.  
The use of surrogate-based optimization lessened the burden of evaluating many design points using the high-fidelity model. A total of 16 high-fidelity model evaluations can be seen in Figure 6a, which is a fraction of the total number of possible discrete design points (341 for the considered values of \(x_{1}\) and \(x_{2}\)). Surrogate-based optimization reduced the computation time by a factor of about 21 based on a convergence criterion of 0.5 \({}^{\circ}\)C if it was assumed that model simulations at all the remaining design points would converge in comparable times. Considering that the 16 high-fidelity model evaluations were completed in 32 h on a machine with an Intel Core i5-3230M CPU at 2.60 GHz using two cores in one socket with 8 GB of RAM, up to 650 h or 95% of computation time were saved. This highlights the benefits of surrogate-based optimization, with the applicability reducing computational requirements of even more complex objective functions such as those within Section 4.3 that consider UFC age distributions via shape functions.  
Looking at functions that can adequately approximate given objective, black-box functions would be ideal surrogate candidates to employ with surrogate-based optimization as the most reductions in model evaluations, and computation time would be attained due to accelerated convergence. The performance of the cubic polynomial surrogate function in Figure 6a can be compared against other polynomial and RBF surrogate functions in Figure 11 of Appendix A. Of the investigated surrogate functions, the fewest iterations were required when using the cubic polynomial surrogate function when starting from the same initial design points. For the given design case of an inventory with identical UFC age, the thin-plate spline RBF surrogate function would have been a poor choice because it required a total of 28 high-fidelity model evaluations--nearly double the 16 evaluations, i.e., double the computation time, required when using the cubic polynomial surrogate function.