## 6 Conclusions  
In this research, we presented a DFO framework that allowed for direct search methods and model-based methods to be united into a single algorithm. The DQL framework showed advantages over other methods in the literature. First, unlike heuristic based methods, convergence was mathematically proven under reasonable assumptions. Second, unlike more rigid DFO methods, the DQL framework was flexible, allowing the combination of direct search, quadratic step, and linear step methods into a single algorithm. This balance of mathematical rigour and algorithmic flexibility created a framework with a high potential for future use. The algorithm was further examined numerically. In particular, we benchmarked the developed DQL method's strategy combinations to determine the optimal combination. The benchmark implied that there was no obvious winner. This motivated the development of the Smart DQL method. We presented the pseudocode for the Smart DQL method and conducted an additional benchmark. The Smart DQL method outperformed all other DQL methods in the benchmark. Last, the Smart DQL method was used to solve the solid-tank design problem. The Smart DQL method was able to produce higher-quality solutions for this real-world application compared to the DQL method, which verified the high performance of the decision-making mechanism. While the DQL and Smart DQL methods both balanced mathematical rigour and algorithmic flexibility, it is worth noting their drawbacks. The most notable one is that the \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \(x_{bl}\) & \(x_{bc}\) & \(x_{lp}\) & \(x_{ma}\) & \(x_{be}\) & \(x_{be2}\) & \(x_{ecc2}\) & \(x_{ds}\) \\
**Profile** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & & **(mm)** & & **(mm)** \\ \hline Water & 122.6 & \(-\)4.3 & 94.0 & 79.8 & 0.8 & 70.0 & 0 & 23.4 \\ FlexyDos3D & 114.0 & 0 & 100.0 & 68.3 & 0.1 & 70.0 & 0 & 0.5 \\ ClearViewTM & 114.0 & 0 & 100.0 & 93.3 & 1.0 & 70.8 & 0.3 & 46.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Optima for Solid-Tank Design Problem (\(n=8\)). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dimension** & **Method** & **Water** & **FlexyDos3D** & **ClearViewTM** \\  & & \(\mathbf{r^{1}=1.3316}\) & \(\mathbf{r=1.4225}\) & \(\mathbf{r=1.3447}\) \\ \hline \(n=5\) & DQL & 0.801 & 0.979 & 0.952 \\ \(n=5\) & Smart DQL & 0.829 & 0.981 & 0.956 \\ \(n=8\) & DQL & 0.767 & 0.977 & 0.686 \\ \(n=8\) & Smart DQL & 0.857 & 0.974 & 0.831 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Optimum Scores for Solid-Tank Design Problem \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(x_{bl}\) & \(x_{bc}\) & \(x_{lp}\) & \(x_{ma}\) & \(x_{be2}\) & \(x_{ecc2}\) & \(x_{ds}\) \\
**Profile** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & & **(mm)** & & **(mm)** \\ \hline Water & 252.4 & 19.2 & 71.8 & 70.1 & 0 \\ FlexyDos3D & 282.0 & 5.8 & 51.8 & 67.0 & 0 \\ ClearViewTM & 225.1 & 21.2 & 63.1 & 69.0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Optima for Solid-Tank Design Problem (\(n=5\)). implementations of DQL and Smart DQL are both at the prototype stage. In comparison to more mature implementations (such as that of SID-PSM [13]), DQL and Smart DQL are unlikely to compete at this time. Another drawback is the need to asymptotically focus on direct search to ensure convergence. Further study will work to advance the Smart DQL method both in the quality of its implementation and the requirements for convergence. Conceptualization, method, validation, and writing and editing: [PERSON] and [PERSON]; first draft: [PERSON]; revisions: [PERSON] and [PERSON]; DQL and Smart DQL software: [PERSON]; solid-tank software: A.O. ; discussion and insight: [PERSON], [PERSON], A.J., A.O., S.C., and M.H. ; supervision and funding: [PERSON] and [PERSON]. All authors have read and agreed to the published version of the manuscript. This research was funded by NSERC, Discovery Grant #2018-03865, and the University of British Columbia. Not applicable. The authors declare no conflict of interest.