## 3 Problem Definition and Formulation
Figure 1: The layout of a distribution center where each picker is allocated to a single buffer: B0—start buffer; B7—out buffer; B1–B6—order picking buffers; Dark colors in buffers B1–B6—principal buffer products; light colors in buffers B1–B6—products also allocated in other buffers. Figure 3: The layout of a distribution center where more than one picker is allocated to one buffer: B0—start buffer; B7—out buffer; B1–B6—order picking buffers; Dark colors in buffers B1–B6—principal buffer products; light colors in buffers B1–B6—products also allocated in other buffers. Figure 1 shows one picker allocated to each buffer and one picker assigned to each buffer. In Figure 2, one picker is allocated to both buffers B4 and B5, making Picker 4 one of the multi-buffer pickers. In Figure 3, more than one picker is allocated to a single buffer; for example, pickers 1, 2, and 5, 6 are assigned to buffers B1 and B6, respectively. In this research, we define the following main mathematical model. Indices and sets:  
\(N\)--number of orders;  
\(H\)--number of products;  
\(R\)--number of buffers;  
\(W\)--number of pickers;  
\(i\), \(m\)--index for orders, \(i\), \(m=1,\ldots,N\);  
\(j\), \(k\)--index for products, \(j,k=1,\ldots,H\);  
\(b\), \(g\)--index for buffers, \(b,g=1,\ldots,R\);  
\(p\), \(l\)--index for pickers, \(p=1,\ldots,W\). Parameters:  
\(t_{ij}\)--picking time of the product \(j\) in the order \(i\);  
\(s_{b}\)--travel time from the depot to the buffer \(b\);  
\(c_{ij}\)--quantity of product \(j\) in the order \(i\) that must be picked by pickers;  
\(q_{bj}\)--available quantity of the product \(j\) in the buffer \(b\);  
\(f_{bg}\)--travel time from the buffer \(b\) to the buffer \(g\) (because the conveyor is one-directional, \(f_{bg}\
eq f_{gb}\));  
\(z_{bj}\)--binary parameter of location products in buffers;  
\(z_{bj}=\left\{ \begin{array}{ll}1,&\mbox{if the product $j$ is located}\\ &\mbox{in the buffer $b$}\\ 0,&\mbox{otherwise}\end{array} \right\}\);  
\(v_{pb}\)--binary parameter of assignment pickers to buffers;  
\(v_{pb}=\left\{ \begin{array}{ll}1,&\mbox{if the picker $p$ is assigned}\\ &\mbox{to the buffer $b$}\\ 0,&\mbox{otherwise}\end{array} \right\}\);  
Decision variables in the main model:  
\(x_{ijb}\)--start time for the product \(j\) in the order \(i\) picked by the picker \(p\) in the buffer \(b\);  
\(y_{ijpb}\)--quantity of the product \(j\) in the order \(i\) picked by the picker \(p\) in the buffer \(b\). Decision variable in the step 1 matheuristics:  
\(y_{ijb}\)--quantity of the product \(j\) in the order \(i\) picked by the picker \(p\) in the buffer \(b\). Decision expressions in the step 1 matheuristics:  
\(t_{p}^{bfr}\)--processing time of the picker \(p\);  
\(t_{b}^{bfr}\)--occupied time of the buffer \(b\). Decision variable the step 2 matheuristics:  
\(x_{ijpb}\)--start time for the product \(j\) in the order \(i\) picked by the picker \(p\) in the buffer \(b\);  
Minimize the makespan:  
\[\min\max_{i=1,\ldots,Nj=1,\ldots,Hp=1,\ldots,Wb=1,\ldots,R}(x_{ijpb}+t_{ij}), \tag{1}\]  
Subject to:  
Pickers to buffers assignment, i.e., the picker does not pick products in buffers to which the picker is not assigned. \[\forall(p)\forall(i)\forall(j)\forall(b:v_{pb}=0)[y_{ijpb}=0], \tag{2}\]  
Products to buffers assignment, i.e., picker does not pick products in buffers in which the product is not located. \[\forall(p)\forall(i)\forall(j)\forall(b:z_{bj}=0)[y_{ijpb}=0], \tag{3}\]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. \[\forall(i)\forall(j,k:j\
eq k,t_{ij}\
eq 0,t_{ik}\
eq 0)[\max_{p=1, \ldots,Wb=1,\ldots,R}(x_{ijpb}+t_{ij})\leq\min_{p=1,\ldots,Wb=1,\ldots,R}x_{ ikpb}\vee\] \[\vee\max_{p=1,\ldots,Wb=1,\ldots,R}\max_{(ikpb+t_{ik})}\leq\min_{p =1,\ldots,Wb=1,\ldots,R}\min_{x_{ijpb}}x_{ijpb}]\], (4)  
Each picker processes just one product for one order in all of his buffers at a time.