## 5 Computational Experiments, Results, and Discussion
### Results for Large-Sized Instances  
The purpose of the computational experiments conducted using large-sized instances was to compare the performance of both MOSS and NSGA-II at different levels of the problem parameters. The main problem parameters that define its structure include the number of workstations (\(|\mathcal{W}|\)), the number of jobs (\(|\mathcal{J}|\)), the loading level (\(LL\)), the percentage of late-arriving jobs (\(PLJ\)), and the percentage of busy machines (\(PBM\)). As defined in [3], \(LL\) is used during the instance generation to determine whether a job requires processing in a workstation randomly. This is done by generating a uniform random number (\(u\)) between 0 and 1 and comparing it to the value of \(LL\). If \(u<LL\), the job will require processing in the workstation. During the instance-generation procedure, this process is repeated for all jobs and workstations. The value of \(LL\) is between 0 and 1, and the higher the value of \(LL\), the more dense the schedule will be. Since the studied DMOSP is characterized by a subset of jobs with non-zero arrival times, the \(PLJ\) parameter was used in instance generation to account for this characteristic. The value of \(PLJ\) was defined as a decimal number between 0 and 100. During the release time generation of an instance, a uniform random number, \(p\in[0\%,100\%]\), was generated,  
Figure 10: Pareto fronts and generated non-dominated solutions of both NSGA-II and MOSS for selected small-sized instances. (**a**) DMOSP-S-7. (**b**) DMOSP-S-6. (**c**) DMOSP-S-24. (**d**) DMOSP-S-17. (**e**) DMOSP-S-4. (**f**) DMOSP-S-15. and its value was compared to \(PLJ\). Whenever \(p>PLJ\), zero release time was assigned to that job. Otherwise, a non-zero release time was randomly generated. Similarly, a machine can be busy at the beginning of a rescheduling phase and, therefore, have a non-zero available time. This is represented using the \(PBM\) parameter, which is used similarly as \(PLJ\) in the instance-generation process. The chosen levels for the mentioned problem structural parameters are shown in Table 6. All values of the processing times of jobs on machines, job arrival times, and machines' available times were randomly generated for each instance using the procedure described in [3]. Based on a consideration of five structural parameters with two levels each, a \(2^{5-1}_{\mathcal{V}}\) fractional factorial design was used. This experimental design was suitable for the requirements of the current comparison since more than two-factor interactions were not of concern. At each treatment combination, 10 instances were randomly generated. For each instance, there were 30 independent runs conducted using each metaheuristic. This resulted in a total of 4800 independent runs for each metaheuristic. The computational time of a run was limited to \(0.6\times|\mathcal{W}|\times|\mathcal{J}|\times LL\) seconds. For NSGA-II runs, the best parameters determined in [4] were utilized. The algorithmic parameters used for small instances were used for the developed MOSS, except that the value of \(RefSetSize\) was set to 10. As reported earlier in the parameter-tuning experiments, this was not expected to reduce the quality of generated solutions, yet it reduced the computational effort needed. For each randomly generated instance, the average of the 30 \(HV\%(\mathcal{D})\) values obtained via a metaheuristic was calculated and denoted \(\overline{HV\%}\). Accordingly, the difference between the MOSS result (\(\overline{HV\%}_{\textit{MOSS}}\)) and the NSGA-II result (\(\overline{HV\%}_{\textit{NSGA-II}}\)) was denoted and evaluated as \(\Delta\overline{HV\%}=\overline{HV\%}_{\textit{MOSS}}-\overline{HV\%}_{ \textit{NSGA-II}}\). The values of \(\Delta\overline{HV\%}\) at all treatment combinations were used as the response in comparing the results of both metaheuristics. Clearly, positive values of \(\Delta\overline{HV\%}\) indicate the better performance of MOSS over NSGA-II.