## 3 Algorithm Based on Limited Search Space
### Initial Solution  
This section explains the time-based integrated initial solution algorithm. Using this algorithm, an initial solution is generated by assigning appropriate workers and operations to the available machines over time based on several priority rules. Before applying the priority rules, the operations are generated based on the inventory level and batch size as follows. First, the required production quantity is determined by comparing the order quantity and initial inventory level. Second, the operation quantity is determined as a multiple of the batch size at the time bucket with a positive production quantity. Third, the production quantity at the next time bucket is determined when the operation quantity is greater than the production quantity. The second and third steps are repeated as needed, and the same process is applied to the subitems according to the MLPS.  
After generating a list of operations, three types of priority rules are applied for the initial assignment: operation rule, machine rule, and worker rule. First, in the case of the operation rule, a higher priority is randomly assigned to the orders that need to be fulfilled within a day, and a lower priority is assigned to the other orders. Second, according to the machine rule, an available machine with a smaller number of executable items has a higher priority. Figure 3 illustrates an example explaining the machine rule. There are three machines (\(M_{1}\), \(M_{2}\), and \(M_{3}\)) and five items (\(I_{1,1,1}\), \(I_{1,1,2}\), \(I_{1,1,3}\), \(I_{2,0,1}\), and \(I_{3,0,1}\)). All five items are executable on \(M_{1}\), and \(I_{1,1,1}\), \(I_{1,1,2}\) and \(I_{2,0,1}\) are executable on \(M_{2}\). \(M_{3}\) only produces item \(I_{1,1,1}\). Therefore, the number of executable items of \(M_{1}\), \(M_{2}\), and \(M_{3}\) are \(5,3\), and \(1\), respectively. According to the machine rule, \(M_{3}\) has the highest priority; thus, item \(I_{1,1,1}\) is assigned to \(M_{3}\), as depicted in Figure 3a. Then, \(M_{2}\) becomes available with the smallest number of executable items; thus, an item is assigned to \(M_{2}\). In this manner, all the items are assigned to the machines. In contrast, Figure 3b illustrates an example where an available machine with a larger number of executable items has a higher priority. In this case, most of the items are assigned to \(M_{1}\), and \(M_{3}\) becomes idle. This example demonstrates that the proposed machine rule is highly likely to derive a better solution in terms of the idle time, makespan, and workload of the machine. Finally, the worker rule identifies a worker with the highest skill for a machine among the available workers at the beginning of the production. To efficiently derive an initial solution, an available machine is selected based on the machine rule, and an appropriate worker with the highest priority is selected. Then, the item with the highest priority is assigned to the selected machine. Figures 4 and A1 depict the flowchart and pseudocode for the time-based integrated initial solution algorithm.  
Figure 2: Flowcharts of the algorithms. (**a**) Proposed algorithm; (**b**) GA; (**c**) GA-VNS.