# Optimization of Warehouse Operations with Genetic Algorithms
## 5 The Proposed Method
### Data Format and Problem Encoding
As all the p-values in the last column of Table 4 are smaller than 0.05, it can be assumed that all the methods are significantly different from one another. As can be seen in Figures 9 and 10, the best results are obtained for the multiple restart of genetic algorithms with AEX crossover operator (MR-AEX) for product placement optimization (see Section 5.4.3 and Algorithm 1) together genetic algorithm with multi-parent HGreX crossover operator (MP-HGreX) for order picking route optimization (see Figure 4). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Prod. Plc.** & **Route** & **w1** & **w2** & **w3** & **w4** & **w5** & **w6** \\ \hline random & random & 0 & 0 & 0 & 0 & 0 \\ random & N.Neighbor & 1.4 & 2.0 & 2.0 & 1.0 & 1.0 & 1.8 \\ random & HGreX & 3.1 & 4.4 & 3.7 & 2.0 & 2.1 & 3.4 \\ random & MP-HGreX & 3.2 & 4.8 & 3.7 & 2.0 & 2.1 & 3.4 \\ AEX & N.Neighbor & 130 & 204 & 169 & 59 & 77 & 186 \\ AEX & HGreX & 744 & 1329 & 1121 & 423 & 495 & 1077 \\ AEX & MP-HGreX & 762 & 1334 & 1077 & 385 & 450 & 1133 \\ MR-AEX & MP-HGreX & 1441 & 1989 & 1667 & 762 & 827 & 1998 \\ AEX/HGreX, no cache & 3127 & 5678 & 4465 & 1710 & 2077 & 5225 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The real running time of the optimization processes in seconds using a computer with two Xeon X5-2696-v2 CPUs, averaged over 10 optimization runs for the experimental data presented in Table 4 and additionally for AEX/HGreX without cache (see Section 5.4.2). ## 7 Conclusions

Shortening the time of order picking is the most important and most beneficial factor in reducing the costs of operating the warehouse (where typically 60% are the costs are generated by order picking [1]). It can be achieved without significant investment by optimizing the locations for particular products in a warehouse and then determining the fastest order completion routes. As the search space of the solutions is enormous (\(9.3\times 10^{157}\) possible placements of 100 products, \(3.1\times 10^{614}\) of 300 products) the problem cannot be analyzed by brute force methods. Thus we presented a complete, fully automatic system based on genetic algorithms, which due to applying intelligent search allows to find the optimal product placements within minutes or tens of minutes for that size of problem (depending on the computer hardware and process parameters). Even though it is not guaranteed that the optimal solution will be found with genetic algorithms, it is possible to find a very close solution to the optimal one, so that in practice it will not make a significant difference. Figure 10: Comparison of the performance of the presented methods with various improvements (see Section 5.4). On horizontal axis: the optimization method. On vertical axis: the average obtained product placement cost (the lower the better) as percentage of the cost with the random product placement and random routes over the six warehouses with order lists presented in Table 4

Figure 9: Graphical representation of the data from Table 4). On horizontal axis: the optimization methods with various improvements (see Section 5.4). On vertical axis: the product placement cost \(costPP\) (the lower the better) obtained with particular methods for the warehouses w1, w2, w3, w4, w5, w6 with corresponding lists of orders. The presented system takes as inputs the warehouse structure (in the form of partial transition costs) and the list of orders and returns the optimal product placement and corresponding shortest order picking routes. Implementation of such a system can accelerate order picking and thus reduce the warehouse operating costs. This allows to serve more customers by the same number of employees in the same time and thus to further increase the sales and profits. The experiments showed that using the multi-parent HGreX crossover improves the results, while for the AEX crossover adding more parents does not change its efficiency.