# Non-Traditional Layout Design for Robotic Mobile Fulfillment System with Multiple Workstations
## 5 Numerical Experiments
### Results in a Traditional Warehouse Layout  
We first observe the location of workstations in the optimal solution when given different numbers of workstations and different length-to-width ratio in traditional layout. Then we will try to produce some rules of thumb to determine the workstation locations that are simple to use while without a loss of accuracy. As part of the experimental results, the optimal solutions for instances when \(N=32\) are shown in Figure 7; more layout figures can be referenced in Appendix A Figure A1. The black solid blocks represent the workstations and each square represents a pod. Using the 4-workstation case as an example, it is interesting to see that the workstations on the top and bottom sides are not symmetric; in other words, the quadrangle consisting of the four workstations is not a rectangle, but a parallelogram, which is counter-intuitive. As the number of workstations increases, the shape has the tendency to turn from a parallelogram to a rectangle. We keep the warehouse length as 40 while varying the width. The optimal workstation layout is illustrated in Figure 8. As the width of the warehouse inc \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{\(N\)} & \multirow{2}{*}{**Length**} & \multicolumn{2}{c}{**Distance Calculation Time(s)**} \\ \cline{3-4}  & & **Traditional** & **Flying-V** \\ \hline \multirow{4}{*}{20} & 60 & 6.50 & 41.65 \\ \cline{2-4}  & 80 & 8.90 & 79.17 \\ \cline{2-4}  & 100 & 11.38 & 111.35 \\ \cline{2-4}  & 120 & 13.70 & 158.33 \\ \hline \multirow{4}{*}{30} & 90 & 21.31 & 473.15 \\ \cline{2-4}  & 120 & 28.94 & 748.92 \\ \cline{2-4}  & 150 & 36.41 & 9647.43 \\ \cline{2-4}  & 180 & 44.84 & 30,384.77 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Running time comparison. Figure 7: Optimal workstation layout under traditional layout. nature of the workstations becomes more obvious; that is, the workstation layout changes from that of a rectangle to a parallelogram. As it would be time-consuming to calculate the distance matrix and acquire the optimal solution by solving the proposed workstation location model (i.e., Equations (18)-(22), which is NP-hard) for medium and large-scale warehouses, even though it is tractable, we herein propose some general design rules by which the workstations can be easily deployed. We perform numerical experiments to verify the validity of these rules. If the number of workstations to be located on the top (or the bottom) is \(n\), the width (in meters) of the warehouse is \(w\), then we have two rules:  
* **rule 1**: \(2n\) rule. By this rule, the edge of the warehouse is equally divided by points with interval of \(\frac{w}{2n}\) and workstations are inserted at the dividing points with odd index. * **rule 2**: \(n+1\) rule. By this rule, workstations are inserted at the dividing points with equal interval of \(\frac{w}{n+1}\). Figure 9 demonstrates examples of rules 1 and 2 with different numbers of workstations, and compares them with the optimal workstation locations. To evaluate the proposed two rules, we conduct two set of experiments, with an odd and even number of workstations, respectively, to compare the travel distance obtained by the proposed optimization model, the \(2n\) rule and the \(n+1\) rule. The results are reported in Tables 5 and 6, in which the GAP represent the percentage of extra travel distance compared with the optimal solution. Both tables show that the two rules are effective, because even the maximum Gap is around 5%. Between the two rules, the \(2n\) rule exhibits better performance with a Gap less than 1.5%. Please note that if the resultant workstations are not located at the end of a picking aisle according to the rules, we will adjust them to a location nearby. Figure 8: Optimal workstation layout under different \(N\) (i.e., number of columns).