# AMR-Assisted Order Picking: Models for Picker-to-Parts Systems in a Two-Blocks Warehouse
## 4 Implementations of an AMR-Assisted Picker-to-Parts System
### Separated Handover Locations for Sub-Aisles  
In the second implementation considered, and presented below, the same handover location is shared between the two pickers operating in a pair of sub-aisles, as depicted in Figure 3. The AMR can visit each handover location twice, either while moving forward or backward in the corridor. As in the implementation described above in Section 4.1, the AMR can eventually turn back to the depot when the right-most aisle containing an order is reached. A pair of opposite sub-aisles is associated with the same handover location. There is a total of \(n\) pairs (\(n+1\) pairs, if the one consisting of the depot instances is also considered). The special feature of this configuration is that the picker of each sub-aisle has the option of loading the AMR either when it moves towards the right-most corridor, or in its way back towards the depot. This choice is possible because, as previously anticipated, according to this warehouse structure, the pair of aisles share the same handover location. Note that this implementation adds complexity and requires more synchronization than the one discussed in Section 4.1, but provides greater flexibility, both in the coordination between different operators and in the coordination between the operator and the cart. This will hopefully lead to more efficient solutions. A Mixed Integer Linear Programming Formulation  
Concerning the mathematical model formulation of this second two-block warehouse configuration, the differences from model described in Section 4.1.1 are highlighted below. The numbering of the sub-corridors is done clockwise. The set \(Y\) contains the pairs of sub-corridors that are associated with the same handover location and are denoted by  
Figure 3: Warehouse with a two-blocks layout and shared handover locations. and \(y(i)\), respectively. For each corridor \(i\) belonging to the set \(V\), the associated corridor \(y(i)\) in \(Y\) in the pair is calculated as:  
\[y(i)=2n\ -\ i+1\]  
where \(n\) is the number of sub-corridors per block and, consequently, handover locations. This formula is also valid for depot instances \(0\) and \(2n+1\). Finally, the binary variable \(q_{bi}\) takes different values, depending on whether the operator of a corridor \(i\) loads the items on the AMR in correspondence to the visit for its corridor or in that associated to corridor \(y(i)\). For the sake of clarity, we present the model in its integrity, although some of the constraints had been preciously introduced already. We also remind the reader that a summary of the notation used can be found in Box 1. \[min\ \sum_{o\ \in\ O}t_{o} \tag{15}\]  
\[\sum_{b\ \in\ B}x_{abk}\leq 1\ \forall a\in A;\ \forall k\in K \tag{16}\]  
\[\sum_{k\in K}\sum_{a\in A}x_{abk}\ =\ 1\ \forall\ b\ \in\ B \tag{17}\]  
\[z_{ibd}+z_{idb}=1\ \forall b,\ d\in B;\ b\
eq d;\ \forall i\in C_{b};\ i\in C_{d} \tag{18}\]  
\[h_{bi}+l_{bi}-M\cdot(1-z_{ibd})\leq s_{di}\ \forall\ b,\ d\ \in\ B;\ b\ \
eq\ d;\ \forall\ i\ \in\ Cb;\ i\ \in\ Cd \tag{19}\]  
\[q_{bi}=1\ \ \forall\ b\ \in\ B;\ i=0;\ i=2n+1 \tag{20}\]  
\[s_{bi}+p_{bi}\leq h_{bi}\ \forall\ b\ \in\ B;\ \forall\ i\ \in\ Cb \tag{21}\]  
Figure 3:0  
Figure 3:1Figure 3:2  
Figure 3:3  
The objective of minimizing the total tardiness of all customer orders is defined in Constraint (15). Constraints (16) and (17) guarantee that the AMR processes the batches one at time. Constraint (18) define the order according to which the batches are to be processed by order picker of sub-aisle \(i\in V\). Constraint (19) define the order picker's start time of picking the batch items stored in her/his picking sub-aisle. Constraint (20) guarantee that the AMR must start and finish its travel in the depot. Constraints (21)-(23) guarantees that the order picker cannot hand over the picked items to the appropriate AMR until (i) she/he has returned to the handover location with these items (see Constraint (21)), and (ii) the AMR has arrived at the associated handover location (see Constraints (22) and (23)).