# 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
\begin{cases}l_{p,a}+l_{p,S_{3}}^{a},&\text{if $(i,j)\in\text{ area $A$}$}\\ \min\Bigl{\{}l_{p,a}+l_{p,S_{3}}^{a},\quad l_{p}^{top}+x_{ij}+x_{\text{max}}+l_{S _{3}}^{top}\Bigr{\}},&\text{if $(i,j)\in\text{area $B$}$}\\ \min\Bigl{\{}l_{p,a}+l_{p,S_{1}}^{a}+l_{a},\quad l_{p}^{top}+x_{ij}+x_{\text{ max}}+l_{S_{3}}^{top}\Bigr{\}},&\text{if $(i,j)\in\text{area $C$}$}\\ \min\Bigl{\{}l_{p,a}+l_{p,S_{1}}^{a}+l_{a},\quad l_{p}^{bot}+x_{ij}+l_{S_{3}}^ {top}\Bigr{\}},&\text{if $(i,j)\in\text{area $D$}$}\\ \end{cases}  \tag{11}\]  
Similarly, we have  
\[d_{ij}^{S_{4}}= \begin{cases}\min\Bigl{\{}l_{p,a}+l_{p,S_{1}}^{a}+l_{a},\quad l _{p}^{bot}+x_{ij}+l_{S_{4}}^{top}\Bigr{\}},&\text{if $(i,j)\in\text{area $A$}$}\\ \min\Bigl{\{}l_{p,a}+l_{p,S_{1}}^{a}+l_{a},\quad l_{p}^{top}+x_{ij}+x_{\text{ max}}+l_{S_{4}}^{top}\Bigr{\}},&\text{if $(i,j)\in\text{area $B$}$}\\ \min\Bigl{\{}l_{p,a}+l_{p,S_{4}}^{a},\quad l_{p}^{top}+x_{ij}+x_{\text{max}}+l _{S_{4}}^{top}\Bigr{\}},&\text{if $(i,j)\in\text{area $C$}$}\\ l_{p,a}+l_{p,S_{4}}^{a},&\text{if $(i,j)\in\text{area $D$}$}\\ \end{cases}  \tag{12}\]  
For other cases,  
\[d_{ij}^{A_{i}}= \begin{cases}l_{p}^{bot}+|x_{ij}-x_{A_{i}}|,&\text{if $(i,j)\in\text{ area $A$}$}\\ l_{p,a}+l_{p,A_{i}}^{a}+l_{p,A_{i}}^{bot},&\text{if $(i,j)\in\text{area $B1$}$}\\ l_{p}^{bot}+|x_{ij}-x_{A_{i}}|,&\text{if $(i,j)\in\text{area $B2$}$}\\ l_{p,a}+l_{p,S_{1}}^{a}+|x_{A_{i}}|,&\text{if $(i,j)\in\text{area $C$}$}\\ \min\Bigl{\{}l_{p,a}+l_{p,S_{1}}^{a}+|x_{A_{i}}|,\quad l_{p}^{bot}+|x_{ij}-x_ {A_{i}}|\Bigr{\}},&\text{if $(i,j)\in\text{area $D$}$}\\ \end{cases}  \tag{13}\]  
\[d_{ij}^{D_{i}}= \begin{cases}\min\Bigl{\{}l_{p,a}+l_{p,S_{1}}^{a}+|x_{D_{1}}|, \quad l_{p}^{bot}+|x_{ij}-x_{D_{1}}|\Bigr{\}},&\text{if $(i,j)\in\text{area $A$}$}\\ l_{p,a}+l_{p,S_{1}}^{a}+|x_{D_{1}}|,&\text{if $(i,j)\in\text{area $B$}$}\\ l_{p}^{bot}+|x_{ij}-x_{D_{1}}|,&\text{if $(i,j)\in\text{area $C1$}$}\\ l_{p,a}+l_{p,D_{1}}^{a}+l_{p,D_{1}}^{bot},&\text{if $(i,j)\in\text{area $C2$}$}\\ l_{p}^{bot}+|x_{ij}-x_{D_{i}}|,&\text{if $(i,j)\in\text{area $D$}$}\\ \end{cases}  \tag{14}\]  
\[d_{ij}^{B_{i}}= \begin{cases}l_{p}^{top}+|x_{ij}-x_{B_{i}}|,&\text{if $(i,j)\in\text{area $A1$}$}\\ l_{p,a}+l_{p,B_{i}}^{a}+l_{p,B_{i}}^{a},&\text{if $(i,j)\in\text{area $A2$}$}\\ l_{p}^{top}+|x_{ij}-x_{B_{i}}|,&\text{if $(i,j)\in\text{area $B,C$}$}\\ l_{p,a}+l_{p,D_{1}}^{a}+l_{p,D_{1}}^{bot},&\text{if $(i,j)\in\text{area $C2$}$}\\ \min\{l_{p}^{bot}+x_{ij}+l_{p,B_{i}}^{a}+l_{p,B_{i}}^{a},\quad l_{p,a}+l_{p, S_{1}}^{a}+l_{S_{1},B_{i}}^{a}+l_{p,B_{i}}^{a,top},\\ l_{p}^{top}+|x_{ij}-x_{B_{i}}|\},&\text{if $(i,j)\in\text{area $D$}$}\\ \end{cases}  \tag{15}\]  
\[d_{ij}^{C_{i}}= \begin{cases}\min\{l_{p}^{bot}+x_{ij}+l_{p,C_{i}}^{a}+l_{p,C_{ i}}^{a,top},\\ l_{p,a}+l_{p,S_{1}}^{a}+l_{S_{1},C_{i}}^{a,top}+l_{p,C_{i}}^{a,top},l_{p}^{top}+|x_{ ij}-x_{C_{i}}|\},&\text{if $(i,j)\in\text{area $A$}$}\\ l_{p}^{top}+|x_{ij}-x_{C_{i}}|,&\text{if $(i,j)\in\text{area $B,C$}$}\\ \end{cases}  \tag{16}\]  
Suppose the selected workstation set is \(\mathcal{R}^{*}\) in the optimal layout, where \(|\mathcal{R}^{*}|=K\), and \(\mathcal{R}^{*}\subseteq\mathcal{R}\), then any pod \((i,j)\in\mathcal{J}\) will be assigned to the workstation with shortesttravel distance in optimal solution (this conclusion can be easily derived according to Equations (18)-(22)), i.e., the actual travel distance of this pod in optimal solution must be  
\[d_{ij}^{\text{actual}}=\min_{q\in\mathcal{R}^{*}}\left\{d_{ij}^{q}\right\} \tag{17}\]