## 5 Experimental Results
### Complexity Analysis of the MILP Model  
Model complexity directly determines the ability of the mathematical model to process data for computational solution. After verifying the feasibility of the MILP model, it  
Figure 6: The Gantt chart of the optimal schedule (Makespan = 63 s). is necessary to analyze its model complexity. Generally speaking, model complexity encompasses two aspects: size complexity and computation complexity. Regarding the size complexity of the MILP model formulated in Section 4.2, three evaluation indices are utilized: (i) the number of binary decision variables (NBV), (ii) the number of constraints (NC) and (iii) the number of continuous decision variables (NCV). Regarding the computation complexity, two evaluation indices are employed: (iv) the optimal production cycle (FV) and (v) the CPU time (Times). The computation complexity generally depends on the algorithm, which determines the ability of the model to efficiently and accurately identify the optimal solution. All of these five evaluation indices can be obtained easily and directly from the property box of the CPLEX interface. The solution time of CPLEX is set to 600 s, and the corresponding experimental results are shown in Table 1.  
Note that, except for the first three test instances, none of the other test instances could obtain the optimal solution/schedule within the prefixed period. Therefore, the FV values from the test are actually the result (i.e., the sub-optimal solution/schedule) when CPLEX is set to 600 s.  
As can be seen in Table 1, the size complexity of the MILP model varies with the test instances. Specifically, the NBV, NC and NCV indices tend to non-linearly increase with the size growth of the test instances, with NC and NCV presenting the largest and smallest increase rates, respectively. As illustrated by the test instances J3-S2-M3-A2 and J6-S2-M3-A2, when the number of jobs increases by 1.00, the NBV, NC and NCV indices increase by 3.17, 3.61 and 2, respectively. As demonstrated in Table 1, the first three test instances can rapidly and precisely determine the optimal solution within a relatively brief period, whereas the last three test instances cannot obtain the corresponding optimal target value of the CPLEX within 600 s. For example, instance J9-S4-M5-A2 fails to obtain a feasible sub-optimal solution, which is denoted by the \"-\" symbol. This phenomenon indicates that as the problem/instance size increases, the difficulty of solving the model increases exponentially, which is reflected in the escalation of computation complexity. In fact, the size complexity index, i.e., NBV, exhibits a strong correlation with the computation complexity index, i.e., Times, of the formulated MILP model. As indicated in Table 1, instances like J6-S2-M3-A2, where NBV is below 1200, can be solved within 10 min, suggesting the model's suitability for daily scheduling in SMEs. When handling larger-scale problems, we need to consider the following approaches: (1) Setting time limits: According to the actual demand, CPLEX's solution/time limit can be flexibly set. Increasing it for large-scale problems allows the model more time to find better solutions, but it also requires a balance between the time cost and solution quality; (2) Model simplification and decomposition: By simplifying the model or decomposing it into smaller sub-problems for separate solutions and coordination, we can reduce its complexity; (3) Proof-based approximations: Advance termination when the optimality \begin{table}
\begin{tabular}{c c c c c c} \hline
**Test Instances** & **NBV** & **NC** & **NCV** & **FV** & **Times** \\ \hline J3-S2-M3-A2 & 138 & 553 & 31 & 38 & 0.33 \\ J3-S4-M5-A2 & 348 & 2187 & 61 & 91 & 4.25 \\ J6-S2-M3-A2 & 438 & 1999 & 61 & 55 & 7.68 \\ J6-S4-M5-A2 & 1164 & 8322 & 121 & 132 & 600 \\ J9-S2-M3-A2 & 900 & 4345 & 91 & 76 & 600 \\ J9-S4-M5-A2 & 2488 & 18417 & 181 & - & 600 \\ \hline \end{tabular} Note. “-” indicates failure to obtain a feasible solution within 600 s.  
\end{table}