# Optimization of Warehouse Operations with Genetic Algorithms
## 5 The Proposed Method
### Data Format and Problem Encoding
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Prod. Plc.** & **Rout** & **Cost** & **w1** & **w2** & **w3** & **w4** & **w5** & **w6** & **Cost** & \(t\)**-Test** \\
**Optimiz.** & **Optimiz.** & & & & & & & & **vs. Rnd.** & **Wilcox.** \\ \hline random & random & average & 29,346 & 49,374 & 42,942 & 14,156 & 16,934 & 40109 & 1.000 & \\  & & std.dev. & 5445 & 9447 & 9050 & 2872 & 3747 & 7792 & 0.207 & 0.0001 \\ \hline random & N.Neighbor & average & 16,931 & 32,879 & 27,501 & 9325 & 9102 & 20314 & 0.596 & 0.0001 \\  & & std.dev. & 2399 & 4444 & 3686 & 1354 & 1220 & 2915 & 0.083 & 0.0104 \\ \hline random & HGreX & average & 14,889 & 31,382 & 26,543 & 8950 & 8080 & 19060 & 0.575 & 0.00001 \\  & & std.dev. & 2238 & 4139 & 3246 & 1047 & 995 & 2328 & 0.071 & 0.0102 \\ \hline random & MP-HGreX & average & 14,342 & 30,660 & 26,004 & 8206 & 7917 & 18667 & 0.544 & 0.0001 \\  & & std.dev. & 1529 & 3074 & 2042 & 684 & 712 & 1529 & 0.048 & 0.0001 \\ \hline AEX & N.Neighbor & average & 6398 & 11,198 & 10,039 & 7274 & 3278 & 7400 & 0.262 & 0.00001 \\  & & std.dev. & 418 & 747 & 619 & 525 & 212 & 498 & 0.018 & 0.0001 \\ \hline AEX & HGreX & average & 5989 & 10760 & 9516 & 6048 & 3160 & 6898 & 0.238 & 0.0001 \\  & & std.dev. & 550 & 809 & 826 & 453 & 253 & 544 & 0.019 & 0.0038 \\ \hline AEX & MP-HGreX & average & 5694 & 10,767 & 9003 & 6000 & 3164 & 6778 & 0.227 & 0.00026 \\  & & std.dev. & 382 & 649 & 533 & 403 & 221 & 441 & 0.015 & 0.0011 \\ \hline MR-AEX & MP-HGreX & average & 5394 & 9676 & 8543 & 5312 & 3023 & 6428 & 0.213 & 0.00001 \\  & & std.dev. & 171 & 339 & 289 & 174 & 101 & 226 & 0.007 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The obtained product placement cost (the lower the better) as the sum of all order picking routes (see Equation (2)) for product placement and order picking route optimization methods with various improvements (see Section 5.4) for the six sample warehouses: w1, w2, w3, w4, w5, w6 with corresponding lists of orders, averaged over 10 optimization runs. The running times are presented in Table 5.

Table 4 presents the detailed results for six sample warehouse structures and order lists (this is the maximum number of warehouses, which can fit in one row of the table). MP-HGreX stands for Multi-Parent HGreX with 8 parents, MR-AEX is Multiple-Restart AEX with 5 restarts, saving the population after 10 iterations, and then continuing with the population of the best individual (see Section 5.4.3 and Figure 2 for explanations). Table 5 presents the real running times of the optimization processes (including I/O operations and Dijkstra Algorithm). The possible reduction of product placement cost depends on the character of the orders. The biggest improvement due to route optimization can be obtained for the orders containing long list of products. The highest improvement due to product placement optimization can be achieved when the orders frequently contain products of particular groups and the frequency with which particular products appear in the orders differs a lot. Thus the improvement possible to achieve is determined mostly by the properties of the orders. Thus, particular methods must be compared among each other for the same warehouse structure and for the same list of orders (This is similar, like in classification, where the possible accuracy depends on the dataset properties and various classifiers must be compared on the same data). The _cost vs. rnd._ column in Table 4 contains the average relative reduction of the product placement cost calculated as _cost vs. rnd._\(=Average(Sum(F1p(w)/random(w))\), where \(w=1...6\) is the warehouse number. The last column contains statistical significance tests calculated on the whole data between two adjacent methods and therefore it is printed in-between rows of the compared methods. Since some persons prefer the T-test and others the Wilcoxon Signed Rank Test for this kind of data, we used both tests to satisfy everyone.