# Two-Stage Metaheuristic Algorithms for Order-Batching and Routing Problems
## 4 Proposed Algorithm
### Initial Batch Order  
Through order consolidation, each reduced batch can shorten the picking route of an order. Consequently, the only factor considered in determining the initial batch is capacity restriction. Moreover, we excluded route proximity from the criteria used to determine  
Figure 3: The framework of the proposed algorithm. the initial batch. We considered five approaches for obtaining the initial solutions of the batching problem, and among these approaches, the picking route was not considered. The single-order (SO) method differs from the other methods, and this method involves considering one order as one picking batch. The random batch (RB) method was adopted to enable comparison between random and planned operations. [PERSON] and [PERSON] [54] focused on a one-dimensional bin-packing problem and proposed three evaluation methods to minimize the number of batches: the first-fit-decreasing (FFD), best-fit (BF), and best-fit-decreasing (BFD) approaches. Because the bin-packing problem is similar to the order-batching problem, we thus adopted these three approaches in this study. The FFD method is applied to address weight limitations. Because of the capacity restriction for each batch, on the basis of the FFD method, orders with heavy weights should be assigned first; however, when such orders are assigned later, they generate a new batch, thus increasing the total number of batches and adding another picking route. The batches can be determined on the basis of the principle of determining the average batch weight (i.e., the BF method). We eventually combined the FFD and BF methods and denoted the integrated method as BFD. In this hybrid method, orders are assigned initially according to the FFD method and their weight, and they are subsequently assigned according to the BF method. The SO method was adopted to enable comparison of the results before and after batch consolidation. In contrast to batching methods, which are used to consolidate several orders into a single order, the SO method facilitates arranging the order-picking sequence. Each order comprises a batch, and dissimilar orders are categorized into different batches. The RB method involves randomly generating each order's assigned batch. The orders are randomly selected from the pool of orders and assigned to a batch individually. When the batch to which an order is assigned cannot accommodate the weight of the order, the system reassigns another batch. The process continues until all orders are assigned. FFD method  
Compared with light-weight orders, heavy-weight orders tend to be more difficult to assign to batches. Therefore, the system starts assigning batches from the heaviest orders. In addition, each batch load is expected to reach the maximum load. The orders are assigned to batches sequentially according to their weight, from the heaviest to the lightest. The assigning principle involves minimizing the remaining weight of each batch. When a batch is fully loaded, the order is reallocated to a new batch. BF method  
Maintaining a balanced batch weight facilitates neighboring solutions to jump from the local optima when the swap move operator is used. The assignment starts from the heaviest order, and an order is assigned to the batch with the greatest remaining weight. In other words, when there are several batches to select from, the system selects the batch with the lowest remaining capacity (batch total capacity-batch present capacity) after an order is assigned to it. Specifically, an order is assigned to the least full batch, and when the batch cannot contain the order, the order is reassigned to a new batch, thus balancing the weight among batches. BFD method  
This method is a combination of the BF and FFD methods. Orders are first prioritized using the FFD approach and are subsequently assigned to batches through the BF method. Orders are assigned sequentially from the heaviest to the lightest batch, and during the assignment, the guiding principle involves minimizing the remaining weight of a batch.