# 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
#### 4.1.1 A Mixed Integer Linear Programming Formulation
\[min\ \sum_{o\in\ O}t_{o} \tag{1}\]  
\[\sum_{b\in\ B}x_{abk}\leq 1\ \forall a\in A;\ \forall k\in K \tag{2}\]  
\[\sum_{k\in K}\sum_{a\in A}x_{abk}\ =1\ \forall\ b\in B \tag{3}\]  
\[z_{ibd}+z_{idb}=1\ \forall\ b,d\in B;\ b\ \
eq\ d;\ \forall i\in C_{b};\ i\in C _{d} \tag{4}\]  
\[h_{bi}+l_{bi}-M\cdot(1-z_{ibd})\leq s_{di}\ \forall\ b,d\in B;\ b\ \
eq\ d;\ \forall i\in C_{b};\ i\in C_{d} \tag{5}\]  
\[s_{bi}+p_{bi}\leq h_{bi}\ \forall\ b\in B;\ \forall i\in C_{b} \tag{6}\]  
\[a_{bi}\leq h_{bi}\ \forall\ b\in B;\ \forall i\in C_{b} \tag{7}\]  
\[a_{bj}\geq h_{bi}+l_{bi}+t_{ij}\ \forall\ b\in B;\forall i\in C_{b}\backslash\{n +1\};\ j\in C_{b}\backslash\{0\};i<j \tag{8}\]  
\[ct_{ak}\ \geq\ ct_{ak-1}+h_{bi}\ +l_{bi}+u_{b}+t_{i,n+1}-M\cdot(1-x_{abk})\ \forall\ a\in A;\ \forall\ b\in B;\ \forall\ i\in C_{b}\{n+1\};\ \forall k\in K \backslash\{1\} \tag{9}\]  
\[ct_{a1}\ \geq\ h_{bi}\ +l_{bi}+u_{b}+t_{i,n+1}-M\cdot(1-x_{ab1}) \tag{10}\]  
\[\sum_{b\in\ B}x_{abk}\leq 1\ \forall a\in A;\ \forall k\in K \tag{2}\]0  
\[\sum_{b\in\ B}x_{abk}\leq 1\ \forall a\in A;\ \forall k\in K \tag{2}\]1  
\[\sum_{b\in\ B}x_{abk}\leq 1\ \forall a\in A;\ \forall k\in K \tag{2}\]2  
\[\sum_{b\in\ B}x_{abk}\leq 1\ \forall a\in A;\ \forall k\in K \tag{2}\]3The objective of minimizing the total tardiness of all customer orders is defined in Constraint (1). Constraints (2) and (3) guarantee that the AMR processes the batches one at time. Constraints (4) define the order according to which the batches are to be processed by the order picker of each sub-aisle. Constraint (5) define the order picker's start time of picking the batch items stored in her/ picking sub-aisle. Constraints (6) and (7) 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 (6)), and (ii) the AMR has arrived at her/his handover location (see Constraint (7)). Constraint (8) determine the arrival time of the AMR handling batch \(b\in B\) at each handover location \(j\in V\). Inequalities (9) guarantee that two batches do not overlap and ensure feasibility with respect to time. Inequalities (10) determine the completion time of the first batch of each AMR. Constraint (11) link the arrival time of the AMR handling batch \(d\in B\) at the first relevant handover location \(i\in C_{d}\) to the completion time of batch \(b\in B\) (plus the time the AMR requires to travel from the depot to handover location \(i\in C_{d}\)) if the following holds: (i) batch \(b\in B\) is handled by the same AMR as batch \(d\in B\), and (ii) batch \(b\in B\) is processed before batch \(d\in B\) by the AMR. Constraint (12) calculate the tardiness for each customer order. Finally, the continuous decision variables and the binary decision variables are defined in Constraints (13) and (14), respectively. The model presented can be algorithmically solved by black-box optimizers in short times (see Section 5). Once the optimal assignment of the variables of the model is found, the optimal operational scheduling for the pickers can be derived as follows. The sequence of picking actions requested to the worker operating in sub-aisle \(i\) are obtained from the values of the timing variables \(s_{bi}\), that contain for each batch \(b\) the (eventual) start of the picking process. It is straightforward to derive a user-friendly list for each picker starting from this information. Combining the optimal value of the set of variables \(x\) and \(ct\) it is finally possible to derive the sequence of operations carried out by the AMRs and to consequently program them.