## 1 Introduction
### Motivational Case Study  
The warehouse system of interest includes a reserve area, forward area, and buffer area. The internal replenishment process in this system is carried out using cages with constant batch sizes. On the other hand, the reserve area consists of narrow corridors that accommodate the bulk storage of all parts. This area is equipped with specialized machines capable of retrieving large cages that carry spare parts, which are not suitable for order picking. Therefore, it is only preferred that parts from this area be picked up if required. In certain instances of replenishment from reserve to forward, the cage is added to the existing inventory on the shelf, potentially exceeding its capacity. In such cases, the warehouse personnel use buffer areas instead of returning the excess inventory to the reserve area. Warehouse managers determine these buffer areas arbitrarily, and they are located next to the shelves without any physical designation as special zones. Their primary purpose is to prevent flow obstruction in the reserve area by avoiding the transportation of partially filled cages. However, if these areas are not effectively managed, they can accumulate a significant amount of stock, leading to congestion in the warehouse. To address this issue, the prioritization of this area is emphasized during the order picking process. Consequently, when inventory is present in both the buffer area and the shelves of the forward area, pickers are instructed to take the stock in the buffer area first. Figure 1 shows the overview of the system.  
The findings reveal that incorporating a strategic buffer area policy within the forward-reserve replenishment framework significantly enhances warehouse operations. Key results include a dramatic reduction in direct picking activities up to 90% in some scenarios alongside optimized buffer area utilization and decreased handling costs. The use of demand forecasts, combined with the integration of ADP, enables the system to dynamically adapt to fluctuating demand, enhancing replenishment efficiency even under labor constraints. By showcasing these benefits, this study presents a scalable and adaptable model for optimizing warehouse operations in high-demand, multi-product environments.  
This paper makes an essential contribution to the forward reserve area problem and reveals that internal replenishment processes can be carried out more smoothly with a well-managed buffer area.  
The specific objectives of the work in this paper can be further described as follows:  
* To overcome the computational challenges of in-warehouse replenishment processes and progressively improving policies by adopting an ADP approach with certainty equivalent.
* To evaluate the impact of the ADP method in managing intuitively placed buffer points for internal replenishment and develop strategies to alleviate warehouse congestion.
* To demonstrate that internal replenishment processes can be carried out more smoothly in a well-managed buffer area.  
This paper is structured as follows: Section 2 reviews the relevant literature on the forward-reserve problem and identifies existing gaps. Section 3 details the methodology, including the ADP framework and the MILP model. Section 4 presents the experimental results from different scenarios and analyzes the model's performance. Section 5 discusses managerial insights, limitations, and future research directions. Finally, Section 6 provides concluding remarks.