## 6 Experimental Simulations
### Picking
#### 6.1.2 The TSP Solution  
As mentioned in Section 3.2, the Traveling Salesman Problem can be adapted to the problem we study by treating the shopping cart as the vehicle and the zones to visit as the customers: zone 1 represents the \"start depot\" as it contains the entry while zone 15 represents the \"end depot\" since the shopping bags are stored here when waiting to be picked up. The set \(V\) of vertices that makes up the graph \(G\) is a subset of the zones set that contains zone 1, zone 15, and the zones from which at least one product needs to be picked up. Since with this solution each zone must be visited only once, all the products in the same zone should be grouped together and collected when arriving at each zone. For example, order 8 contains products from the following zones: 1, 2, 9, 8, 3, 6, and 15 (1 and 15 are included by default as they are the start and end points). The zones are highlighted on the layout in Figure 2. The edges are constructed according to the vertices contained in \(V\), and the respective cost associated with each edge can be retrieved from the Origin-Destination travel time matrix in Table 1 in Section 3.1. This operation is represented  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{Order 1} & \multicolumn{2}{c}{Order 2} & \multicolumn{2}{c}{Order 3} & \multicolumn{2}{c}{Order 4} & \multicolumn{2}{c}{Order 5} \\ \hline \(t_{ij}\) & Zone & \(t_{ij}\) & Zone & \(t_{ij}\) & Zone & \(t_{ij}\) & Zone & \(t_{ij}\) & Zone \\ \hline / & 1 & / & 1 & / & 1 & / & 1 & / & 1 \\
51.26 & 11 & 51.26 & 11 & 51.26 & 11 & 51.26 & 11 & 51.26 & 11 \\
25.92 & 12 & 25.92 & 12 & 22.04 & 12 & 25.92 & 12 & 25.92 & 12 \\
29.46 & 10 & 29.46 & 10 & 29.46 & 10 & 29.46 & 10 & 29.46 & 10 \\
29.46 & 12 & 16.47 & 11 & 16.47 & 11 & 29.46 & 12 & 16.47 & 11 \\
29.46 & 10 & 16.47 & 10 & 16.47 & 10 & 29.46 & 10 & 16.47 & 10 \\
29.46 & 12 & 16.47 & 11 & 29.46 & 12 & 29.46 & 12 & 16.47 & 11 \\
29.46 & 10 & 16.47 & 10 & 29.46 & 10 & 29.46 & 10 & 25.92 & 12 \\
27.06 & 6 & 29.46 & 12 & 29.46 & 12 & 29.46 & 12 & 29.46 & 10 \\
68.28 & 15 & 29.46 & 12 & 16.47 & 11 & 38.82 & 3 & 29.46 & 10 \\  & & 29.46 & 10 & 42.35 & 3 & 68.28 & 15 & 16.47 & 11 \\  & & 27.06 & 6 & 68.28 & 15 & & & 42.35 & 3 \\  & & 31.76 & 3 & & & & & 28.24 & 7 \\  & & 28.24 & 7 & & & & & 28.24 & 3 \\  & & 52.98 & 15 & & & & & 68.28 & 15 \\ \hline
351.58 s & 430.40 s & 380.63 s & 390.50 s & 453.92 s & \\ \hline \multicolumn{2}{c}{Order 6} & \multicolumn{2}{c}{Order 7} & \multicolumn{2}{c}{Order 8} & \multicolumn{2}{c}{Order 9} & \multicolumn{2}{c}{Order 10} \\ \hline \(t_{ij}\) & Zone & \(t_{ij}\) & Zone & \(t_{ij}\) & Zone & \(t_{ij}\) & Zone & \(t_{ij}\) & Zone \\ \hline / & 1 & / & 1 & / & 1 & / & 1 & / & 1 \\
51.26 & 11 & 51.26 & 11 & 15.58 & 2 & 51.26 & 11 & 44.20 & 9 \\
25.92 & 12 & 16.47 & 10 & 28.62 & 9 & 22.04 & 12 & 16.47 & 8 \\
25.92 & 11 & 29.46 & 12 & 16.47 & 8 & 29.46 & 10 & 25.09 & 2 \\
25.92 & 12 & 25.92 & 11 & 16.47 & 9 & 16.47 & 11 & 28.62 & 9 \\
25.92 & 11 & 35.68 & 2 & 35.29 & 3 & 25.92 & 12 & 20.00 & 11 \\
16.47 & 10 & 24.26 & 3 & 35.29 & 9 & 29.46 & 10 & 42.35 & 3 \\
16.47 & 11 & 68.28 & 15 & 16.47 & 8 & 29.46 & 12 & 42.35 & 11 \\
16.47 & 10 & & & 31.76 & 3 & 29.46 & 10 & 23.53 & 8 \\
29.46 & 12 & & &partly in Figure 2 for better understanding. The shopping list is rearranged according to the TSP optimal solution that gives the lowest travel time between the zones. The travel times for the BOPS orders when the TSP optimization is implemented are shown in Table 3. It is a coincidence that all lists are sorted in ascending order, due to how the zones are numbered.