## 4 Surrogate-Based Optimization
### Optimal Arrangements for Inventories with Assorted UFC Ages
#### 4.3.2 Kumaraswamy PDF-Based Shape Function  
In contrast to those shown in Figure 7, the results of the surrogate-based optimization in Figure 8 varied with different values of the shape parameter \(a\) when UFC ages were arranged using a Kumaraswamy PDF-based shape function. Thus, for a given room spacing, the UFC spacing necessary can be quite different depending on how one wishes to arrange UFCs by age. By using a shape function and parameterizing the UFC age arrangement, the UFC spacings required for many arrangements (within one family of functions) were able to be studied. To our knowledge, this method has not yet been used in the design process of the DGR.  
Figure 7: Optimal design points for Equation (1) at specified room spacings when UFCs with ages of 30–60 years are arranged using a cosine-based shape function, as determined from the surrogate-based optimization of Equation (2) using a cubic RBF surrogate function. (**a**) Room spacing of 10 m, (**b**) Room spacing of 15 m, (**c**) Room spacing of 20 m, (**d**) Room spacing of 25 m.  
Based on the nature of the 100 \({}^{\circ}\)C contour in Figure 8b, the choice of cubic RBF surrogate functions was justified because polynomial surrogates would not have been able to produce such a curve. This can be seen if Figure 8b is compared to Figure A2 of Appendix A, which shows the results from using a cubic polynomial surrogate function.  
The results in Figure 8 demonstrate that the largest UFC spacings are required for shape parameter values of \(a\to 0.25\) and \(a\to 4.00\), while smaller UFC spacings are allowed for shape parameter values in between those values. These results agree with the age distribution examples depicted in Figure 5. Shape parameter values \(a\to 0.25\) and \(a\to 4.00\) would place the greatest number of young, high heat-generating UFCs together (at the start/end and middle of the placement rooms, respectively); thus, more distance between the containers would be required to keep the temperature around them manageable.  
For this age arrangement at a room spacing of 10 m, only shape parameter values \(0.75\leq a\leq 2.50\) can lead to maximum temperatures below 100 \({}^{\circ}\)C (Figure 8a). From another perspective, Figure 8 shows that the largest required UFC spacings should have age arrangements that follow a Kumaraswamy PDF-based shape function. At room spacings of 15, 20, and 25 m, the largest required UFC spacings are 1.6, 1.3, and 1.1 m, respectively. This can be concluded as shape parameter values of \(a\to 0.25\) and \(a\to 4.00\) most unfavorably place UFCs by age. Should there be deviations from the age arrangements specified by  
Figure 8: Optimal design points for Equation (1) at specified room spacings when UFCs with ages 30–60 years are arranged using a Kumaraswamy-based shape function, as determined from the surrogate-based optimization of Equation (2) using a cubic RBF surrogate function. (**a**) Room spacing of 10 m, (**b**) Room spacing of 15 m, (**c**) Room spacing of 20 m, (**d**) Room spacing of 25 m.  
these values, the young and high heat-generating UFCs can only be further dispersed from their alike peers.