# Non-Traditional Layout Design for Robotic Mobile Fulfillment System with Multiple Workstations
## 5 Numerical Experiments
### Results on Flying-V Warehouse Layout
#### 5.4.2 The Influence of the Cross-Aisle Angle on Total Travel Distance  
When there is only one workstation in the warehouse with a flying-V layout, the total distance will decrease as the cross-aisle angle increases, until the cross aisles gradually close \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\theta\)} & \multicolumn{6}{c}{\(K=4\)} & \multicolumn{6}{c}{\(K=6\)} \\ \cline{2-10}  & \(TD_{op}^{a}\) & \(TD_{2n}^{b}\) & **gap(\%)** & \(TD_{n+1}^{c}\) & **gap(\%)** & \(TD_{op}\) & \(TD_{2n}\) & **gap(\%)** & \(TD_{n+1}\) & **gap(\%)** \\ \hline \(\frac{5}{36}\tau\) & 71,593.30 & 74,140.09 & 3.56 & 74,373.93 & 3.88 & 63,549.64 & 66,542.36 & 4.71 & 65,326.33 & 2.80 \\ \(\frac{7}{36}\tau\) & 70,067.83 & 73,469.44 & 4.85 & 71,725.79 & 2.37 & 61,665.91 & 65,606.88 & 6.39 & 63,191.80 & 2.47 \\ \(\frac{9}{36}\tau\) & 69,477.01 & 73,148.87 & 5.28 & 69,932.52 & 0.66 & 60,795.30 & 64,901.33 & 6.75 & 60,897.85 & 0.17 \\ \(\frac{11}{36}\tau\) & 69,594.93 & 72,503.00 & 4.18 & 69,594.93 & 0.00 & 61,283.73 & 64,252.63 & 4.84 & 62,465.60 & 1.93 \\ \(\frac{13}{36}\tau\) & 68,592.55 & 68,668.83 & 0.11 & 70,926.76 & 3.40 & 60,709.21 & 61,344.47 & 1.05 & 65,064.70 & 7.17 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Evaluation of two rules for a flying-V layout with varying angles and even workstation numbers. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\mathbf{L}\)} & \multicolumn{6}{c}{\(K=3\)} & \multicolumn{6}{c}{\(K=5\)} \\ \cline{2-10}  & \(TD_{op}^{a}\) & \(TD_{2n}^{b}\) & **gap(\%)** & \(TD_{n+1}^{c}\) & **gap(\%)** & \(TD_{op}\) & \(TD_{2n}\) & **gap(\%)** & \(TD_{n+1}\) & **gap(\%)** \\ \hline
30 & 15,817.00 & 16,231.29 & 2.62 & 17,014.56 & 7.57 & 12,341.79 & 12,405.33 & 0.51 & 14,198.08 & 15.04 \\ \hline
40 & 25,211.04 & 25,239.64 & 0.11 & 26,032.61 & 3.26 & 19,942.51 & 19,942.51 & 0.00 & 20,725.88 & 3.93 \\ \hline
50 & 35,804.80 & 35,816.23 & 0.03 & 36,453.01 & 1.81 & 28,742.93 & 29,407.60 & 2.31 & 28,955.16 & 0.74 \\ \hline
60 & 47,627.74 & 47,627.74 & 0.00 & 48,242.22 & 1.29 & 38,656.56 & 40,590.70 & 5.00 & 38,788.78 & 0.34 \\ \hline
70 & 60,928.13 & 60,928.13 & 0.00 & 61,622.61 & 1.14 & 50,167.67 & 53,375.80 & 6.39 & 50,222.41 & 0.11 \\ \hline
80 & 75,828.51 & 75,828.51 & 0.00 & 76,602.99 & 1.02 & 63,236.30 & 67,760.89 & 7.16 & 63,256.03 & 0.03 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Evaluation of two rules for a flying-V layout with varying lengths and odd workstation numbers. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\theta\)} & \multicolumn{6}{c}{\(K=3\)} & \multicolumn{6}{c}{\(K=5\)} \\ \cline{2-10}  & \(TD_{op}^{a}\) & \(TD_{2n}^{b}\) & **gap(\%)** & \(TD_{n+1}^{c}\) & **gap(\%)** & \(TD_{op}\) & \(TD_{2n}\) & **gap(\%)** & \(TD_{n+1}\) & **gap(\%)** \\ \hline
30 & 13,930.99 & 14,281.08 & 2.51 & 15,106.58 & 8.44 & 11,647.16 & 11,807.75 & 1.38 & 13,439.90 & 15.39 \\ \hline
40 & 22,452.60 & 22,542.24 & 0.40 & 22,562.43 & 0.49 & 18,725.86 & 18,922.95 & 1.05 & 19,647.70 & 4.92 \\ \hline
50 & 31,848.53 & 32,581.00 & 2.30 & 31,848.53 & 0.00 & 27,351.13 & 27,828.04 & 1.74 & 27,556.98 & 0.75 \\ \hline
60 & 42,905.27 & 44,448.00 & 3.60 & 42,905.27 & 0.00 & 37,070.60 & 38,531.14 & 3.94 & 37,070.60 & 0.00 \\ \hline
70 & 55,470.00 & 58,023.00 & 4.60 & 55,618.90 & 0.27 & 48,164.98 & 50,916.24 & 5.71 & 48,184.23 & 0.04 \\ \hline
80 & 69,477.01 & 73,148.87 & 5.28 & 69,932.52 & 0.66 & 60,795.30 & 64,901.33 & 6.75 & 60,897.85 & 0.17 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Evaluation of two rules for a flying-V layout with varying lengths and even workstation numbers.