## 3 Methodology
### MILP Model for Forward Area Replenishment Problem with Buffer Area
_Model_:  
\[min\sum_{i=1,t=1}^{|N|,T}Q_{it}c^{r}+C_{it}^{D}c^{d}+(WAR_{it}-DW_{it}+WBR_{it} )c^{w}+X_{it}^{W}c^{nxx} \tag{1}\]  
_Subject to:_  
\[XAR_{it}=XBR_{it}+QN_{it}C_{i}-X_{it}^{W} i\in N,\,1\leq t\leq T \tag{2}\]  
\[XBR_{it+1}=XAR_{it}-D_{it}^{S} i\in N,\,1\leq t\leq T \tag{3}\]  
\[WAR_{it}=X_{it}^{W}+WBR_{it} i\in N,\,1\leq t\leq T \tag{4}\]\[WBR_{it+1}=WAR_{it}-D_{it}^{W}+C_{it}^{D}C_{i}-D_{it}^{P} i\in N,\,1\leq t\leq T \tag{5}\] \[X_{it}{}^{W}\leq Q_{it}C_{i} i\in N,\,1\leq t\leq T\] (6) \[QN_{it}\leq Q_{it}M_{i} i\in N,\,1\leq t\leq T\] (7) \[QN_{it}C_{i}+XB_{it}-X_{it}{}^{W}\leq F_{i} i\in N,\,1\leq t\leq T\] (8) \[D_{it}^{W}+D_{it}^{S}+D_{it}^{P}=D_{it} i\in N,\,1\leq t\leq T\] (9) \[X_{it}{}^{W}\leq XBR_{it} i\in N,\,1\leq t\leq T\] (10) \[XAR_{it}=XBR_{it}+QN_{it}C_{i}-X_{it}^{W} i\in N,\,1\leq t\leq T \tag{2}\]0 (11) \[XAR_{it}=XBR_{it}+QN_{it}C_{i}-X_{it}^{W} i\in N,\,1\leq t\leq T \tag{2}\]1 (12) \[XAR_{it}=XBR_{it}+QN_{it}C_{i}-X_{it}^{W} i\in N,\,1\leq t\leq T \tag{2}\]2 (13) \[XAR_{it}=XBR_{it}+QN_{it}C_{i}-X_{it}^{W} i\in N,\,1\leq t\leq T \tag{2}\]3 (14) \[XAR_{it}=XBR_{it}+QN_{it}C_{i}-X_{it}^{W} i\in N,\,1\leq t\leq T \tag{2}\]4 (15) \[XAR_{it}=XBR_{it}+QN_{it}C_{i}-X_{it}^{W} i\in N,\,1\leq t\leq T \tag{2}\]5 (16) \[XAR_{it}=XBR_{it}+QN_{it}C_{i}-X_{it}^{W} i\in N,\,1\leq t\leq T \tag{2}\]6  
The objective function (1) minimizes overall costs, which include the costs associated with internal replenishment (\(c^{r}\)), direct picking (\(c^{d}\)), inventory holding in the buffer area (\(c^{nv}\)), and transferring inventory from the forward area to the buffer area (\(c^{nv}\)). Based on Assumption 7, the cost of transferring inventory from the forward area to the buffer area is incorporated to prevent unnecessary handling activities. In this model, the cost structure is defined with the assumption that direct picking incurs the highest cost, followed by forward-to-buffer area transfer, then buffer area usage, with replenishment having the lowest cost. This hierarchy reflects the labor intensity and resource requirements of each operation. Constraints (2) and (3) determine the stock levels in the forward area based on the inventory from the preceding time period and the replenishment quantity within the current timeframe. Constraints (4) and (5) model the inventory movement between the buffer and forward areas. Constraints (6) and (7) are the either-or constraints that help the variables \(X_{it}{}^{W},QN_{it}\) and the binary variable \(Q_{it}\) to be assigned correctly. In Equation (18), the replenishment decision is also shown. This binary decision is critical as it triggers the sequence of subsequent inventory adjustments and movements. \[Q_{it}= \begin{cases}1,&\quad\quad\quad\quad\quad\quad\quad\quad\quad if\,\, forward\,\,area\,\,is\,\,replenished\\ 0,&\quad\quad\quad\quad\quad\quad if\,\,forward\,\,area\,\,is\,\,not\,replenished \end{cases}  \tag{18}\]  
Equation (19) shows the process of stock relocation from the forward area to the buffer area within a warehouse. If the transfer meets certain capacity conditions, the binary variable is set to 1, indicating replenishment. If no stock is moved, the variable is set to 0. \[Q_{it}= \begin{cases}1,&\quad\quad\quad\quad\quad\quad\quad if\,C_{i}\geq X_{it }{}^{W}>0\\ 0,&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad if\,X_{it}{}^{W}=0\end{cases}  \tag{19}\]  
Function (20) shows the logic of the inclusion of \(QN_{it}\) is particularly important as it integrates the replenishment decision with the physical quantity of inventory moved, ensuring that our model reflects the practical constraints of the warehouse's operational capacity. Together, these equations fine-tune our inventory management and enhance the overall efficiency of warehouse operations, leading to a reasonable balance between availability and the cost implications of inventory holding and movement. \[Q_{it}=