## 6 Case Study
### Development of a Rule-Based Heuristic Algorithm  
From this case study, certain decision rules can be derived for the storage of incoming parts and for the transshipment of parts between the internal and external warehouse for the specific industrial use case. An overview of the derived decision rules can be found in Figure 9.  
Figure 9 shows that the decision on proactive transshipments depends on the situation if parts have to be transshipped reactively in the period to satisfy demand. If parts have to be transshipped reactively and the internal warehouse has free capacity at the same time, the parts with the highest approximated future transshipment costs in the external warehouse are also transshipped from the external to the internal warehouse. If the internal warehouse is fully utilized, parts can be exchanged. Parts in the internal warehouse with lower approximated future transshipment costs in the external warehouse are exchanged with parts in the external warehouse with higher approximated future transshipment costs in the external warehouse through transshipments. These decision rules based on the decision tree in Figure 9 were programmed in Python as a rule-based heuristic algorithm and applied to the case study data. The pseudocode is provided in Appendix A. Table 9 compares the result of the mixed-integer linear programming model with the result of the rule-based heuristic algorithm.  
\begin{table}
\begin{tabular}{c c} \hline
**Reactive Approach** & **Proactive Approach** \\ _O (\(\sigma\)) Computation Time [s]_ & _O (\(\sigma\)) Computation Time [s]_ \\ \hline
1.3 (0.2) & 1.6 (0.2) \\ \hline \end{tabular}
\end{table}
Table 8: Overview of the average computation time per period and standard deviation.  
\begin{table}
\begin{tabular}{c c} \hline
**Incoming parts** & Transshipments \\ Reactive transshipment necessary to satisfy demand \\ \hline
**Yes** & No \\ \hline
**Transship** & Internal warehouse is fully utilized \\ \hline
**Transship** & Yes & No \\ incoming parts to the internal warehouse & Transship incoming parts to the external warehouse & Transship incoming parts to the internal warehouse to the external warehouse \\ \hline \end{tabular}
\end{table}
Table 9: Comparison of the total cost in different proactive approaches.  
Figure 9: Identification of decision rules from the case study.  
It is interesting to see that the costs with the rule-based heuristic algorithm are equal to the optimal solution of the mixed integer linear programming model. However, when analyzing the results, it was noticed that different decisions were made and therefore different costs resulted in 4 out of 135 periods. This result can be explained by the partially equal demand probabilities of different parts. If the remaining capacity from reactive transports is used for further proactive transshipments, it is possible that not all parts with the same demand probability can be transshipped. Consequently, in both models, it is random which parts with the same demand probability are transshipped. As a result, a part was demanded in a period of the MILP that was not transshipped in the previous period due to the described problem. In contrast to the heuristic, this made another reactive transshipment necessary. On the other hand, reactive transshipments were also avoided in the MILP compared to the heuristic in other periods.  
Basically, the comparison of the mixed-integer linear programming model with the rule-based heuristic algorithm has shown that the derived decision rules implemented in the heuristic match the optimal solution of the mixed-integer linear programming model.