# Compensating Data Shortages in Manufacturing with Monotonicity Knowledge
## 3 Applications in Manufacturing
### Forming and Press Hardening of Sheet Metal  
Another application example is press hardening [47]. In the experimental setup considered here, a blank is placed in a chamber furnace with a furnace temperature \(T_{\text{f}}\) above 900 \({}^{\circ}\)C. After heating the blank, an industrial robot transports it with handling time \(t_{\text{h}}\) into the cooled forming tool. In the following, the extra handling time \(\Delta t_{\text{h}}=t_{\text{h}}-10\,\text{s}\) is used instead, with \(10\,\text{s}\) being the minimum time the robot takes to move the blank from the furnace to the press. The final combined forming and quenching step allows for the variation of the press force \(F_{\text{P}}\) and the quenching time \(t_{\text{q}}\). Afterwards, the formed part is transferred by the industrial robot to a deposition table for further cooling. An illustration of the process chain is shown in Figure 2.  
The goal of the press hardening process considered in this work is to obtain a formed metal part with a prescribed hardness level, where the hardness is measured in units of the Vickers hardness number (unit symbol HV). In order to achieve such a pre-defined hardness, the process operator has to find suitable combinations of the four process parameters \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & **Min** & **Max** & **Phys. Unit** \\ \hline \(T_{\text{f}}\) & 480 & 560 & \({}^{\circ}\)C \\ \(n_{\text{c}}\) & 40 & 50 & — \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Ranges for the process variables of laser glass bending. Figure 1: Side view of the laser glass bending process. Symbols: \(T_{\text{f}}\)—furnace temperature, \(\Delta_{\text{l}}\)—distance between the left- and right-most laser line, \(\beta\)—bending angle. Lengths are given in mm. \(T_{\rm f},\Delta t_{\rm h},F_{\rm p},t_{\rm q}\). And for that purpose, in turn, an appropriate model is needed for the hardness \(y\) of the formed part (at distinguished measurement points on the surface of the part) as a function of the process variables  
\[\mathbf{x}:=(x_{1},\ldots,x_{4}):=(T_{\rm f},\Delta t_{\rm h},F_{\rm p},t_{\rm q}) \in X, \tag{17}\]  
where \(X\subset\mathbb{R}^{4}\) is the hypercuboid set with the bounds specified in Table 2. In particular, such a model has to allow sufficiently accurate real-time predictions in order to help the process operator in quickly searching the parameter space \(X\) for optimal process parameter settings. Since SIAMOR models are polynomial by construction, they perfectly satisfy this real-time requirement. In contrast, due to the four-dimensional parameter space, already a single evaluation of a representative finite-element simulation model of the press-hardening process is prohibitively time-consuming to be of any practical use in quickly exploring the parameter space. As in the case of glass bending, experiments for the press hardening process are expensive because they usually require manual adjustments, which tend to be time-consuming. Additionally, the local hardness measurements at the chosen measurement points on the surface of the quenched part are time-consuming as well. This is why the training data base we used is rather small. It contains 60 points resulting from a design of experiments with the four process variables \(T_{\rm f},\Delta t_{\rm h},F_{\rm p}\) and \(t_{\rm q}\) ranging between the bounds in Table 2, along with the corresponding hardness values at six local measurement points (referred to as MP1,..., MP6 in the following). In order to compensate this data shortage, expert knowledge is brought into play. An expert for press hardening expects the hardness to decrease monotonically with \(\Delta t_{\rm h}\) and to increase monotonically with \(T_{\rm f}\) as well as with \(t_{\rm q}\).