## 4 Matheuristics Development
### Step 2. Order Picking Makespan Minimization Problem (Using \(y_{ijpb}\) as Input from Step 1)  
Using \(y_{ijpb}\) as input from step 1, we define the following mathematical model. From the main model from Section 3, we take the time sequencing constraints. Therefore, constraints (26)-(33) correspond to constraints (4)-(11) from the full model. The goal function (25) corresponds to (1) from the full model. We define only the start time decision variable (34) which corresponds to the decision variable (15). Minimize the makespan:  
\[\min\max_{i=1,\dots,Nj=1,\dots,Hp=1,\dots,Wb=1,\dots,R}(x_{ijpb}+t_{ij}), \tag{25}\]  
Subject to:  
Picking operations have precedence in each order, executed by all pickers in all buffers; i.e., the previous picking operation must finish before the start of any next picking operation for this order. \[\begin{split}\forall(i)\forall(j,k:j\
eq k,t_{ij}\
eq 0,t_{ ik}\
eq 0)[\max_{p=1,\dots,Wb=1,\dots,R}(x_{ijpb}+t_{ij})\leq\min_{p=1,\dots, Wb=1,\dots,R}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!Each picker processes just one product for one order in all of his buffers at a time.  
\[\forall(p)\forall(i,m:i\
eq m)\forall(b:v_{pb}=1)\forall(j)[x_{ijpb}+t_{ij}\leq x _{mipb}\lor x_{mipb}+t_{mj}\leq x_{ijpb}], \tag{27}\]  
Each picker in each order in all his buffers processes just one product at a time. \[\forall(p)\forall(i)\forall(b:v_{pb}=1)\forall(j,k:j\
eq k)[x_{ijpb}+t_{ij} \leq x_{ikpb}\lor x_{ikpb}+t_{ik}\leq x_{ijpb}], \tag{28}\]  
Each picker in each order for all products processes products just at one of his buffers at a time. \[\forall(p)\forall(i)\forall(j)\forall(b,g:b\
eq g,v_{pb}=1,v_{pg}=1)[x_{ijpb }+t_{ij}\leq x_{ippg}\lor x_{ippg}+t_{ij}\leq x_{ijpb}], \tag{29}\]  
Each picker at each of his buffers processes just one of the different orders and just one of the different products at a time. \[\forall(p)\forall(b:v_{pb}=1)\forall(i,m:i\
eq m)\forall(j,k:j\
eq k)[x_{ijpb }+t_{ij}\leq x_{mkpb}\lor x_{mkpb}+t_{mk}\leq x_{ijpb}], \tag{30}\]  
Each picker processes just one of the different jobs, just one of the different products, and just one of  
existed six, four, and eight pickers in the first, second, and third distribution center layouts consequently. Distribution center layouts are shown in Figures 1-3. We generated 120 main and 48 repeated locations in the six buffers. The product was stored in one, two, or three locations in the distribution center. In total, 85 products were stored only in one location. A total of 22 products were stored in two locations. A total of 13 products were stored in three locations. The time limit for CPLEX execution was set at 5 min for each of the solving cases. Table 2 reports the results of the matheuristics' first step solution--pickers free time minimization problem--for five orders. The search was completed, and all instances were solved optimally. The number of solutions varied from one to three for the first layout (instances 1-10), from one to four for the second layout (instances 41-50), and from one to nine for the third layout (instances 81-90). The average time spent solving was 3.73 s and varied from 2.45 s to 13.69 s for the first layout (instances 1-10). The average time spent solving was 2.86 s and varied from 1.33 s to 6.71 s for the second layout (instances 41-50). The average time spent solving was 23.65 s and varied from 4.50 s to 35.96 s for the third layout (instances 81-90). Table 2 also reports the results of the matheuristics' second step solution--order picking makespan minimization problem after solving pickers free time minimization problem--for five orders. The search was completed only for the seventh instance, so only for this instance was an optimal solution found. For 1-10 instances, the average gap to the lower bound was 39.25% and varied from 0.00% to 100.00%. In 41-50 instances, the average gap to the lower bound was 54.17% and varied from 25.56% to 66.93%. In 81-90 instances, the average gap to the lower bound was 63.48% and varied from 39.51% to 100.00%.