# Ant Colony Optimization with Warm-Up
## 4 Computational Experiments
### Results Obtained in the Simulation of the Real Warehouse  
After studying the effect of the proposed warm-up on a set of generic TSP instances, it is in interest of the author to analyze its effect in a more realistic and complex environment. The simulation of a manual warehouse for picking was therefore used, and the proposed ant colony optimization with warm-up is used to define the routing of pickers--i.e., once defined the picking locations the picker has to visit, the order in which they are visited is defined. The faced problem is essentially a TSP, but the graph of nodes and paths is more constrained, with less possible paths between nodes and many mandatory walkways. Importantly, no additional constraints such as capacity of pickers' baskets, definition of batches, or interference between pickers moving through the aisles are considered. Starting from the warehouse layout, a graph of accessible positions is generated placing a node in front of each storage location and a node where aisles cross to each other, and then, using the well-known Floyd-Warshall algorithm, the matrix of minimum distances between nodes is generated. The starting warehouse is made of 20 aisles with 16 storage locations each, crossed by a single cross-aisle in the middle (i.e., between the 8 th and the 9 th locations). Each storage location are 2\(\times\)2 m, aisles are 4 m wide, while the cross-aisle is 8 m wide. The resulting graph used in the tests is shown in Figure 4.  
The results obtained by the compared algorithms in the simulation of the warehouse are reported in Table 5 and can be intuitively visualised looking at Figures 5-7 respectively in terms of _(i)_ cost of the best solution found, _(ii)_ solutions explored before finding the best, and _(iii)_ computational times. The results broadly respect what already seen in previous experiments. On average the proposed ACOWU is still the best in terms of cost even if sometimes it cannot provide a better solution than the classic ACO, but the same could be said for the other algorithms using a pheromone initialization strategy. The Dai algorithm is still in second position and proved to be a very good alternative. Concerning the solutions explored and therefore the computational time Bellaachia algorithm is the best (as already seen in previous experiments). However, the proposed ACOWU is again a good alternative as clearly visible in Figures 6 and 7. Again, as in the previous experiments on generic TSP instances, the difference in terms of solutions explored and computational time is not that big. However, the utilization of a pheromone initialization technique, as already proved in literature, guarantees some advantages over the classic ACO. Figure 3: Comparison of solutions explored on generic TSP instances. Figure 4: Graph of warehouse used for tests. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{8}{c}{**Cost of the Best Solution Found**} \\ \hline \multirow{2}{*}{**N**} & \multicolumn{2}{c}{**ACO**} & \multicolumn{2}{c}{**ACOOWU**} & \multicolumn{2}{c}{**Dai**} & \multicolumn{2}{c}{**Bellaichia**} \\ \cline{2-9}  & **Avg. ** & **St.Dev. ** & **Avg. ** & **St.Dev. ** & **Avg. ** & **St.Dev. ** & **Avg. ** & **St.Dev. ** \\ \hline
20 & 543 & 13 & 552 & 15 & 537 & 9 & 714 & 37 \\
30 & 604 & 14 & 626 & 15 & 616 & 18 & 864 & 34 \\
40 & 804 & 30 & **781** & 18 & 778 & 18 & 1070 & 44 \\
50 & 806 & 30 & **755** & 8 & 792 & 43 & 1138 & 69 \\
60 & 1026 & 144 & **1001** & 56 & 995 & 46 & 1420 & 74 \\ \hline \multicolumn{8}{c}{**Solutions Explored before Finding the Best**} \\ \hline \multirow{2}{*}{**N**} & \multicolumn{2}{c}{**ACO**} & \multicolumn{2}{c}{**ACOWU**} & \multicolumn{2}{c}{**Dai**} & \multicolumn{2}{c}{**Bellaichia**} \\ \cline{2-9}  & **Avg. ** & **St.Dev. ** & **Avg. ** & **St.Dev. ** & **Avg. ** & **St.Dev. ** & **Avg. ** & **St.Dev.