# Two-Stage Metaheuristic Algorithms for Order-Batching and Routing Problems
## 3 Problem Statement  
This study deals with the order-batching problem and picking-route-sequence problem in the picker-to-part system; that is, in the conventional rectangular warehouse, the different orders are first classified and sorted into batches, with group orders with similar paths grouped together as much as possible. For the items to be picked in order, \(O_{k}\), \(i\), \(BO_{l}\), and \(D_{O_{k}/BO_{l}}\) represent the \(k\)-th order, storage location \(i\), \(l\)-th batch, and the pickingdistance of order \(k\) or batch \(l\), respectively. \(m\) represents the distance unit. \(O_{1}=\{3,\ 6,\ 8\}\), indicating that order 1 needs to be picked from storage positions 3, 6, and 8. If there are five orders, \(O_{1}=\{3,\ 6,\ 8\}\); \(O_{2}=\{62,\ 74,\ 80\}\); \(O_{3}=\{2,\ 3,\ 4,\ 6,\ 7\}\); \(O_{4}=\{67,\ 74,\ 80\}\); and \(O_{5}=\{61,\ 62,\ 76,\ 80\}\) are to be classified into batches, and then, the distance of the orders individually calculated. As shown in Figures 1 and 2, if we pick the items by following the number sequence, in order 1, it takes 4 m to travel from the entrance to position 3, 3 m from position 3 to 6, 2 m from position 6 to 8, and 9 m from position 8 to the entrance and exit. Hence, the distances of order 1 to order 5, \(D_{O_{1}}\sim D_{O_{5}}\), are 18 m, 40 m, 16 m, 46 m, and 40 m. Since order 1 and order 3 are both picked in the first aisle, they can be grouped into batch 1, \(BO_{1}=\{2,\ 3,\ 4,\ 6,\ 7,\ 8\}\). The distance of batch 1, \(D_{BO_{1}}\), is 18 m. Without order batching, the total distance of these two orders is 34 m. Orders 2, 4, and 5 can be classified into batch 2, \(BO_{j}=\{61,\ 62,67,74,\ 76,\ 80\}\), based on a similar situation. If the items are picked by individual order, the total distance of these three orders is 126 m, but the distance of batch 2, \(D_{BO_{2}}\), is largely decreased to 46 m.  
Figure 1: Calculation Representation of the Distance. Figure 2: Conventional Warehouse Layout. After combining the orders into batches, the picker picks items in sequence according to the sorted picking list of the batch, but if the picking route is different, the picker might go as far as possible in the warehouse. For example, [37,64,21,19,43] and [19,37,21,43,64] are picking routes 1 and 2 of the order. The distances of picking routes 1 and 2 are 68 m and 42 m, respectively. We adopted the model proposed by [PERSON] et al. [3] to address the batching and picking problems and minimize the overall route. The problem assumptions, notations, and model are as follows. Problem Assumptions:  
1. Warehouses have a picker-to-part system and parallel aisles. 2. The storage strategy is dedicated location, so each storage position has a fixed size. Only one item can be stored or picked, and the quantity of the same item being picked is fixed at one. 3. There is only one picker; that is, there is only one picker on the aisle, and the picker will take the path with the shortest distance and will not choose the path that bypasses the aisle. 4. It is prohibited to split the order into different batches, and the items in the order cannot be cut into different batches. The size of one order cannot be over the size of one batch. 5. The picker can pass up and down the aisle and pick the item on the left or the right. The picker picks the item in front of the storage position, and the distance and time from the picker to the item can be ignored. 6. The warehouse layout plan shown in Figure 2 is a conventional rectangular warehouse, starting from and finally returning to the I/O point to complete the closed loop. There are 10 storage spaces in a row--two rows on the left and right in one aisle, and five aisles, with a total of 100 storage positions. 7. All customers are known in advance, and no orders will be inserted, reduced, or changed in the middle of the process.