## 4 Numerical Experiments
### Evaluation of the Initial Solution Algorithm  
To verify the effectiveness of the initial solution algorithm, we solve small scale problems. The experiment environment is described as follows. * The number of final items is five, and they have the BOM structure in Figure 6. In cases 1 and 2, a BOM can have up to two subparts for the upper part. In cases 3 and 4, the BOM structure is extended up to four levels. * The time horizon is composed of seven time buckets, and the number of finished product s required in each time bucket can be up to 30. * The processing time of each operation is randomly generated within a range from 1 to 15 time units. * The number of machines is five. On an average, each machine can work on processing 30% items, and designated machines are randomly assigned. * Capacity is either low or high. In the case of low capacity, the available time unit per day is 2000, and in the case of high capacity, 4500 units per day are available. * The number of workers is three. The skill level of each worker is randomly generated within a range from 50 to 100%. * Initial inventory level is zero. * The lot size is 30. Tables 1 and 2 show numerical results with simple a BOM structure, as shown in Figure 6a. Order data is randomly generated within a range from 1 to 30, and other data remain the same. The performances of LSFS and LSIS are compared with those of GA, GA-VNS, and LSFS without initial solution algorithm, and LSIS without initial solution algorithm. In addition, the performance measures used are the objective function values (G1, G2, and G3) explained in Section 2 and computation time. Figure 6: Product structure used in numerical experiment. (**a**) The MLPS of Case 1 & 2; (**b**) The MLPS of Case 3 & 4. The results show that LSFS and LSIS with the initial solution algorithm are better than other algorithms. In particular, the performance gap between the algorithm with initial solution and without initial solution algorithm is quite large. As mentioned in the previous section, the main concept of the proposed algorithm is to begin with a relatively good starting point and improve the solution within the limited search space. It is also interesting that the computation time of LSIS and LSFS becomes shorter when the initial solution algorithm is applied. In other words, the performance of LSS-based algorithms is heavily dependent on the initial starting point, and the proposed initial solution algorithm plays a critically important role in achieving good performance. In addition, the performance of LSFS and LSIS is even worse than GA-based algorithms when the initial algorithm is not applied. Different from GA-based algorithms, the proposed algorithm does not use a mutation operator, and it can be worse than GA-based algorithms. The difference in the performances of LSFS and LSIS is not large. However, in terms of computation time, LSIS tends to be slightly superior to LSFS. Next, we consider a more complex BOM structure shown in Figure 6b. We note that the proposed algorithm is terminated after 7200 seconds because it requires more computation time because of the complex BOM structure. Tables 3 and 4 summarize the results. In most cases, LSFS and LSIS perform better than the other algorithms, but GA performs well in case 3 with low capacity. Because of the nature of a small-size problem, in some cases, GA can be better than the strategy limiting the search space and requiring efforts in the local search, which is less efficient than randomly searching for a larger solution space. In both cases, the initial solution algorithm improves the performance of the algorithm.