## 4 Empirical Illustrations  
In the case company, there are four identical parallel injection molding machines that produce plastic pallets. On average, each machine produces 45 pallets per hour. The production manager always schedules the products whose due dates are within about 2 weeks (i.e., 10 working days). Because, at most, one changeover is allowed per day, this research generated nine products randomly for testing. Descriptions of the generation processes are shown in Table 1.  
This research generated 20 sets of data for testing. Both integer program in Phases 1 and 2 are optimized by the commercial software LINGO 17. According to a prior test using a computer with Intel Core i7-9700 3.00 GHz, the maximum run time was set at 2 min for the integer program in Phase 1. As the integer program of Phase 2 is constrained by a makespan generated by Phase 1, longer run times are required. The maximal run time for the integer program of Phase 2 is set as 20 min. In order to reduce the required run time, \(S_{j1}=j\) for all machines are set. The results are shown in Table 2.  
In Table 2, Phase 1 indicates the results generated by the integer program in Phase 1, and Phase 1 + Phase 2 indicates the results generated by the integer program in Phase 2, based on the makespan results from Phase 1. According to the results shown in Table 2, the total tardiness resulting from Phase 1 + Phase 2 are always much shorter than Phase 1, except in Case 20. Therefore, the proposed two-phase methodology can minimize the makespan and total tardiness sequentially. The present study then compared the proposed methodology with the following two heuristics: longest processing time (LPT) and largest weighted slack time (LWST). The LPT is a heuristics that was developed for PMS to minimize the makespan. The unscheduled job with the longest processing time has the highest priority in being assigned to a machine whenever one is free [14]. Regarding the LWST, it incorporates the following two criteria: weight and slack time. For a product with a larger quantity of orders, a 1 day delay will increase the total tardiness more than a product with a smaller quantity of orders. Thus, the products with larger quantities of orders should have higher priority. In this research, the weight of a product was set as the quantity of the received orders, \(A_{i}\). The slack time means the remaining spare time at the current time [20]. The less slack time, the higher the priority of the product. By definition, the slack time is equal to the last day that the model has to be exchanged for producing the product, \(LS_{i}\). Accordingly, LWST arranges all products in decreasing order of \(A_{i}/LS_{i}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Product Data**} & **Random** & **Description** \\  & & **Generation** & **1** \\ \hline \multirow{4}{*}{1} & \multirow{4}{*}{\(PT_{i}\)} & & The number of working days required \\  & & & by a product is between 1 and 5 days. \\  & & [8 h, 110 h] & It can also be found that the \\  & & & production batch sizes of different \\  & & & products are quite different. \\ \hline \multirow{2}{*}{2} & \multirow{2}{*}{\(A_{i}\)} & \(PT_{i}\times\) [10, 40] \(\times\) & The quantity of received orders are \\  & & 0.01 \(\times\) 45 & 10\% to 40\% of the production batches. \\ \hline \multirow{4}{*}{3} & \multirow{4}{*}{\(LS_{i}\)} & & The latest starting day is Day 9, and \\  & & & the maximum production time is \\ \cline{1-1}  & & [Day 1, Day 9] & 5 days. Thus, all products can be \\ \cline{1-1}  & & & completed within 15 days. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The generation of test data. The results for the LPT and LWST are shown in Table 3. It can be seen from the results that with the proposed methodology the makespans are shorter or equal to the LPT and LWST in all cases. For the total tardiness, the proposed methodology had worse results than the LWST in two cases, and the LWST results had longer makespans.