# Non-Traditional Layout Design for Robotic Mobile Fulfillment System with Multiple Workstations
## 3 Problem Description and Distance Expression
### Travel Distance Expression in Flying-V Layout  
Distance equations seem exceedingly complicated under flying-V settings due to the introduction of angled cross-aisle. Although the distance expressions in flying-V layout warehouses with single(double) workstation(s) are provided in several existing literature, e.g., [16; 18; 23; 24; 25; 26; 28], they seem not applicable under multiple workstation cases. As we will elaborate below, the distance equations vary with the area where the pod and targeted workstation are located. To the best of our knowledge, it is the first attempt that the shortest distance equations under flying-V layout with multiple workstations are provided (i.e., Equations (11)-(16) in this section). Figure 3: Sketch of a traditional warehouse layout. In the flying-V layout, two cross aisles are inserted into the warehouse, as shown in Figure 4. The inserted cross aisles begin at the bottom center, extending to the right-most and left-most edges with angle \(\theta\) and \(\pi-\theta\), respectively. Owing to its complexity, we will first demonstrate the expression of robot travel distance with a single workstation located at the center of the bottom edge. There exist two possible candidate paths if we try to deliver pod \((i,j)\) to the workstation \((x_{0},y_{0})\) under a flying-V layout, as shown in Figure 4 (herein, we have \(x_{0}=0,y_{0}=\frac{1}{2}w_{p}\) according to the description in Section 3.1) (Similar descriptions of possible paths in flying-V layout see [50]). Path one (i.e., the solid red line) heads down (or up) from the current position along the corresponding vertical aisle, reaching the angled cross-aisle, and then travels along the angled cross-aisle, and finally arrives at the workstation. Path two, which is identical to the delivery path in a traditional warehouse, moves down along the corresponding vertical aisle, travels horizontally along the bottom aisle, and finally reaches the workstation. We assume that each pod is delivered from its current position to the workstation along the shortest path. Again, we ignore the error caused by the even column index. Thus, the distance of path one can be given by (The exact distance formula of path one is \(\tau_{0}=\frac{1}{2}\big{(}w_{p}+w_{a}\big{)}+\frac{|x_{ij}|-\frac{1}{2}\big{(} w_{p}+w_{a}\big{)}}{\cos\theta}+|\Big{(}|x_{ij}|-\frac{1}{2}\big{(}w_{p}+w_{a} \big{)}\Big{)}\tan\theta-y_{ij}|\). To simplify the calculation, we approximate \(\tau_{0}\) by \(d_{ij}^{p_{1}}=\frac{|x_{ij}|}{\cos\theta}+|\frac{|x_{ij}\tan\theta|}{|x_{ij}| }-y_{ij}|\). Such approximation is reasonable, because the difference between two formulas can be given by \(\epsilon=\tau_{0}-d_{ij}^{p_{1}}=\frac{1}{2}\big{(}w_{p}+w_{a}\big{)}\Big{(} \tan\theta+1-\frac{1}{\cos\theta}\Big{)}\), where \(\theta\in\left[\frac{5}{36}\pi,\frac{13}{36}\pi\right]\), and \(w_{p}=1,w_{a}=2\) in this work. Please note that \(\epsilon\in\left[0.54,1.17\right]\) under the above conditions, which is negligible compared to \(\tau_{0}\), and thus, we replace \(\tau_{0}\) with \(d_{ij}^{p_{1}}\). As for multiple workstation cases, the above conclusion still holds after similar analysis. For other distance calculations, we also adopt similar approximation method, see Table 2):  
\[d_{ij}^{p_{1}}=\frac{|x_{ij}|}{\cos\theta}+||x_{ij}\tan\theta|-y_{ij}|. \tag{5}\]  
Since the trajectory of path two in flying-V warehouses is exactly the same as the delivery path in traditional warehouses, we can derive its distance expression by Equation (4), i.e.,  
\[d_{ij}^{p_{2}} =|x_{ij}-x_{0}|+|y_{ij}-y_{0}|\] \[=|x_{ij}|+y_{ij}-\frac{1}{2}w_{p} \tag{6}\]  
Then the shortest distance from pod \((i,j)\) to workstation \((x_{0},y_{0})\) is  
\[d_{ij}=\min\{d_{ij}^{p_{1}},d_{ij}^{p_{2}}\} \tag{7}\]  
Suppose we need to deploy \(K(1\leqslant K\leqslant|\mathcal{R}|)\) workstations in the warehouse. To facilitate the calculation of distance between any pod and any candidate workstation, we divide the storage region into categories \(A\), \(B\), \(C\) and \(D\) (Figure 5).