## 6 Experimental Simulations
### Picking and Packing  
With the help of the two-step optimization in the previous section, it is possible to estimate the total time required to complete a SOP-based picking and packing operation in a reliable way. With analogous considerations, it is also possible to estimate the time spent at the cashier for the TSP picking and packing operation. Therefore, in this section, we compare the total time cost considering both the picking and packing process as described in Section 5.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & Order 1 & Order 2 & Order 3 & Order 4 & Order 5 \\ \hline No Opt & 351.58 & 430.40 & 380.63 & 390.50 & 453.92 \\ TSP & 170.75 & 183.69 & 150.75 & 150.75 & 163.69 \\ SOP & 348.21 & 444.14 & 403.33 & 344.52 & 488.04 \\ Rel. SOP & 232.38 & 272.56 & 239.62 & 226.68 & 252.56 \\ \hline \hline  & Order 6 & Order 7 & Order 8 & Order 9 & Order 10 \\ \hline No Opt & 583.42 & 251.33 & 481.89 & 508.29 & 466.19 \\ TSP & 183.69 & 150.75 & 141.06 & 183.69 & 154.00 \\ SOP & 504.68 & 399.76 & 440.16 & 445.13 & 490.05 \\ Rel. SOP & 279.62 & 328.98 & 242.17 & 308.37 & 341.83 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison between simulated travel times with different routing strategies implemented for 10 BOPS orders.  
Figure 3: Simulated travel times with different routing strategies implemented for 10 BOPS orders.  
In this simulation, we no longer consider the SOP solution for the reasons given in Section 6.1. Therefore, we consider only three out of five scenarios described in Section 5:  
* TSP + cashier;
* Relaxed SOP + cashier; and
* pick-scan-pack model for Relaxed SOP.  
In all scenarios, 2 s per bag is necessary for preparation. The travel times are derived directly from Section 6.1 for the TSP and Relaxed SOP. Once a zone is reached, each item has an equal probability of being picked, the time to locate and pick the item is estimated as 7 s if it is directly placed in the shopping cart without scanning it or 9 s if the pick-scan-pack model is applied. When the pick-scan-pack model is not implemented, two employees will be occupied at the cashier: the picker and the cashier. The necessary time to complete the loading, scanning, and packing procedure is estimated as 5 s per product for the picker (0.5 to place the items on the conveyor belt, 2.5 for the product to get from the loading point to the scanning point [32], 0.5 to scan the product, and finally 1.5 to place the item into a bag) and 3.5 s per product for the cashier (0.5 for the items to be placed on the conveyor belt, 2.5 for the product to get from the loading point to the scanning point [32], and 0.5 to scan the product). The simulation results are reported in Figure 5.  
From the figure it can be observed that the pick-scan-pack model for Relaxed SOP performs the best. The time difference ranges between 107.78 s and 295.39 s with an average of 202.81 s when compared to the TSP+cashier scenario, and between 279.50 s and 396.50 s with an average of 312.00 s when compared to the Relaxed SOP + cashier scenario. This means that the pick-scan-pack alternative reduces the total time needed by 22.18% and 30.48%. It shows a remarkable improvement that underlines even further the efficiency of the Relaxed SOP optimization when paired with the right instruments (scan-as-you-pick devices) and models (the packing model).  
For the reasons above, the Relaxed SOP is considered the best and most applicable optimization approach with the following advantages:  
* Prevents product damaging without excessively increasing picking time.
* Allows the integration of picking and packing operations.  
Figure 4: Simulation results for 10 order picking and packing tasks.