## 2 Literature Review  
The forward-reserve area problem has been studied repeatedly in the literature. Ref. [2] solved the problem of selecting items to store in automated storage and retrieval systems and allocating space for inventory keeping. They developed a heuristic solution and applied it to a naval supply center's small-sized items where it is assumed that they are continuously divisible. This paper is the first to solve the internal replenishment problem: restocking a primary area from a secondary location in the facility using a mathematical model. The model focuses on maximizing profit by allocating sufficient items in the volume base, considering the replenishment cost. Ref. [3] considered busy and idle periods in a distribution center. The busy period means the order picking operations will occur, and the idle period means replenishment operations of the forward area from the reserve area may happen. They also assumed that one trip from the reserve to the forward area is sufficient, replenishment is performed with one unit load, and order picking from the reserve area is possible at a higher cost. They formulated the model as a binary programming problem that minimizes the total labor time and aims to prevent accidents in busy periods. They solved it with greedy knapsack and dynamic programming. Also, a constraint for limiting replenishment capacity is added, and the computational outcomes are compared. Ref. [4] presented a mathematical model and a heuristic for assigning items and deciding the size of a warehouse's five functional areas: receiving, shipping, forward, reserve, and cross-docking. They mention that determining the sizes of the areas and assigning products to those areas affect warehouse management on strategic, tactical, and  
Figure 1: Internal replenishment process. operational levels. The model is formulated as a mixed-integer linear programming model (MILP) and solved with the proposed heuristic and branch and bound algorithms. The computational results show that the heuristic algorithms take much less time. Ref. [5] applied three storage strategies to the forward area, analyzed the results, and determined which was better. The first one is the equal space strategy, which allocates equivalent space for each item; the other ones are the equal time strategies, which allocate space for each item that is sufficient for the same amount of time, and the optimal strategy. They applied strategies to small parts of the data by extending the model in [2] with new inventory level constraints based on the strategies. Ref. [6] considered a forward reserve area problem in a picker-to-belt system where order pickers collect orders from the forward area and put them on the conveyor. It is assumed that if the pickers face a stockout, the belt stops, and they must perform emergency replenishment. They have compared and evaluated the computational results of replenishment methods, and six heuristic policies have been proposed. Ref. [7] presented the results of early heuristic methodology studies and an alternative branch and bound algorithm that can solve the problem optimally. They applied the algorithms to two different data sets that belong to different warehouses, and the computational performance of the algorithms was presented. It is shown that when the number of stock-keeping units (SKUs) gets smaller, the optimality difference is higher for the aforementioned greedy heuristic in [PERSON]'s study. It is also observed that even though the objective value is equal, the model's outcomes differ between the optimal and the heuristic solutions. Ref. [8] used simulation and what-if analysis to assign items to forward areas and allocate space to them. In order to achieve effective solutions in an acceptable amount of time, several storage policies have been implemented. In earlier studies, the forward pick area was assumed to be continuously divisible; therefore, the models were fluid. Ref.