# Non-Traditional Layout Design for Robotic Mobile Fulfillment System with Multiple Workstations
## 4 The Workstation Location Model  
Section 3 provides the expression of the shortest travel distance from an arbitrary pod dwell location to any candidate workstation. Suppose each pod has the identical possibility of being used for goods retrieval under a random storage policy; it is tractable to solve a location problem to determine the optimal positions of multiple workstations. One may question whether we can enumerate all the possible workstation configurations given the knowledge that number of workstations may be not that large for a single warehouse in practice. To verify this, we set up an experiment using a warehouse with width \(N=32\) and length \(L=30\), 960 pods or about 1500 m\({}^{2}\). Figure 6 shows the total distance with possible workstation combinations for both traditional layout and flying-V layout (\(K=2\)). It takes about one minute of CPU time to compute the the travel distances from each pod to all candidate workstations for traditional layout and four hours for the flying-V layout. If we consider using more workstations, or a larger warehouse, the computation workload will increase sharply. For example, if we select eight workstations  
Figure 6: Total distance for all combinations in the enumeration. in the same warehouse, there are \(C_{34}^{8}\approx 1.8\times 10^{7}\) combinations for the traditional and \(C_{36}^{8}\approx 3\times 10^{7}\) combinations for the flying-V layout. Therefore, the enumeration method cannot work in this case. Based on this information, an easy-to-implement as well as efficient model should be developed to make this decision problem practical. Based on the distance expression developed above, we propose the integer programming model to minimize the total travel distance of robots. This model can determine the number and location of workstations. Please note that for the flying-V layout, we take the angle of the cross aisles as an input parameter, rather than a decision variable. The rationale behind is: (1) making it a decision variable may make the following model intractable; (2) the choices for this angle are limited owing to the discrete nature of pod locations. Let \(\mathcal{R}\) be the set of candidate workstations, \(\mathcal{J}\) the set of all pod storage units (for simplicity, we call them pods), and \(K\) the number of workstations the warehouse operators decide to construct. \(z_{r}\) and \(u_{ijr}\) are binary variables. \(z_{r}\) are called _location variables_, representing whether the candidate point \(r\) is chosen as a workstation in the optimal solution or not. \(u_{ijr}\) are called _allocation variables_, indicating that whether pod \((i,j)\) is assigned to workstation \(r\) or not. Based on the above interpretations, we redescribe the workstation location problem as a facility location problem, a typical NP-hard problem, and formulated it as an integer programming model. The proposed formulation below can be applied under both traditional and flying-V layout. min \[\sum_{(i,j)\in\mathcal{J}}\sum_{r\in\mathcal{R}}d_{ijr}u_{ijr}\] (18) \[\sum_{r\in\mathcal{R}}z_{r}=K,\] (19) \[\sum_{r\in\mathcal{R}}u_{ijr}=1, \forall(i,j)\in\mathcal{J}\] (20) \[\sum_{(i,j)\in\mathcal{J}}u_{ijr}\leqslant z_{r}, \forall r\in\mathcal{R}\] (21) \[z_{r},u_{ijr}\in\{0,1\}, \forall(i,j)\in\mathcal{J},\forall r\in\mathcal{R}.\] (22)  
The objective function Equation (18) aims to minimize the total travel distance from any pod to its assigned workstation along the shortest path as discussed in Section 3. Constraints Equation (19) restricts that exactly \(K\) workstations are selected in optimal solution. Constraints Equation (20) specify that each pod should be assigned to exactly one workstation. Constraints Equation (21) guarantee that no pod will be assigned to candidate workstation \(r\) if it is not selected in optimal solution. Constraints Equation (22) define the variable type.