# Identical Parallel Machine Scheduling Considering Workload Smoothness Index
## 6 Numerical Experiments
### Large Job Instances (\(n\geq 50\))  
Below, we further test the performance of the proposed two heuristic algorithms together with the previous list scheduling algorithms with large job input instances. Especially, we consider two scenarios with regard to the variation range of the processing times of jobs, i.e., \(p_{j}\in[5,10]\) or \(p_{j}\in[8,10]\). Similar to the case of small job input instances, three instances are generated for each combination \((m,n)\), and the average results are reported in Tables 2 and 3, where the running time is in seconds. Notice that in large-scale instances, CPLEX fails to output even feasible solutions within 7200 s for most instances. Due to the better performance of the list algorithm based on the LPT rule compared to that based on SPT and RPT rules in solving small job instances, we set the value of \(obj^{**}\) as the benchmark obtained by \(LSA_{LPT}\), i.e., \(gap_{2}=\frac{obj-obj^{**}}{obj^{**}}\times 100\%\).  
Tables 2 and 3 report the experimental results of \(LSA_{LPT}\), \(LSA_{SPT}\), \(GA_{BM}\) and \(ILSA_{LPT}\). Similar to the small job setting, the results of large job instances in Tables 2 and 3 indicate  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\((m,n)\)} & \multicolumn{2}{c}{\(LSA_{LPT}\)} & \multicolumn{2}{c}{\(LSA_{SPT}\)} & \  
that \(ILSA_{LPT}\) and \(GA_{BM}\) outperform again the other three list algorithms on average in terms of solution quality. \(ILSA_{LPT}\) produces the best solutions for all the test instances, while \(LSA_{LPT}\) and \(GA_{BM}\) output the best schedules for only a part of the instances. Moreover, Tables 2 and 3 report that the average performance of \(LSA_{LPT}\) is worse than \(GA_{BM}\) for the test instances.  
For the solution quality, we observe by Tables 2 and 3 that the objective value of the solution obtained by any of the heuristic algorithms increases proportionally in the number of machines, while the number of jobs has no significant impact on the objective value. Moreover, when the job processing time is scaled in \([8,10]\), the average performance of \(LSA_{RPT}\) is even better than that of \(LSA_{LPT}\). That is to say, compared to small-scale numerical experiments, \(LSA_{LPT}\) lost its power to solve large-scale instances. Similarly, due to the limitations of given iterations and population size, the average performance of the designed genetic algorithm in solving large-scale numerical instances also shows a downward trend. From Tables 2 and 3, we can know that the solution quality of \(GA_{BM}\) is generally inferior to that of \(ILSA_{LPT}\). This is mainly due to the inability of the designed genetic algorithm to fully utilize its performance in solving large job instances under the given chromosome population size and the number of iterations. Moreover, for some combinations where the number of jobs is an integer multiple of the number of machines, the performance of \(GA_{BM}\) is inferior to that of \(LSA_{LPT}\) due to the limitations of the given population size and iterations. Considering the running time, all the heuristic algorithms require running time proportional to the job scale. Meanwhile, the average running time consumed by \(ILSA_{LPT}\) is significantly less than that of \(GA_{BM}\). Actually, the running time of \(GA_{BM}\) is much larger than that of the other four heuristic algorithms.  
In summary, we conclude the following. (1) In general, both \(ILSA_{LPT}\) and \(GA_{BM}\) outperform the previous list scheduling algorithms. (2) For large-scale instances, \(ILSA_{LPT}\) can output the best solutions for all the instances within one second. (3) In large-scale numerical experiments, the list scheduling algorithm based on the LPT rule lost its power to solve the considered problem.