# Identical Parallel Machine Scheduling Considering Workload Smoothness Index
## 6 Numerical Experiments
### Small Job Instances (\(n\leq 15\))  
We first present numerical experiment results for small job input instances, in which the processing time of each job is a positive real number between 5 and 10. For each combination \((m,n)\), we generate three instances with the random seed, and the numerical results in Table 1 are the average objective value of the solutions to the three instances by the corresponding algorithms.  
In Table 1, we observe that the relative error between each of the two proposed heuristic algorithms, i.e., \(ILSA_{LPT}\) and \(GA_{BM}\), and the exact solution is less than 1% in all the small job instances. For the other three previous algorithms, i.e., \(LSA_{RPT}\), \(LSA_{LPT}\) and \(LSA_{SPT}\), their average relative errors are equal to 6.90%, 4.27% and 12.46%, respectively. From the \"\(gap_{1}\)\" column, we observe that the performance of the list algorithm based on the LPT rule is better than that based on SPT and RPT rules. Nevertheless, the list algorithm based on the SPT rule holds the worst performance compared to other list algorithms. There is no significant trend where the relative errors of the tested algorithms increase with the number of either jobs or machines. Specifically, considering that the number of jobs is an integer multiple of that of machines (e.g., the second and fifth combinations), the list scheduling algorithm can output higher-quality feasible solutions. In addition, the average relative errors of \(ILSA_{LPT}\) and \(GA_{BM}\) are about 90.40% and 100.00% smaller than that of  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**(\(m\), \(n\))**} & \multirow{2}{*}{**CPLEX**} & \multicolumn{2}{c}{\(LSA_{RPT}\)} & \multicolumn{2}{c}{\(LSA_{LPT}\)} & \multicolumn{2}{c}{\(LSA_{SPT}\)} & \multicolumn{2}{c}{\(GA_{BM}\)} & \multicolumn{2}{c}{\(ILSA_{LPT}\)} \\ \cline{2-13}  & & \(obj\) & \(gap_{1}\) & \(obj\) & \(gap_{1}\) & \(obj\) & \(gap_{1}\) & \(obj\) & \(gap_{1}\) & \(obj\) & \(gap_{1}\) \\ \hline \((2,5)\) & 6.74 & 7.85 & 16.47 & 7.36 & 9.20 & 8.51 & 26.26 & 6.74 & 0.00 & 6.74 & 0.00 \\ \((2,10)\) & 6.72 & 7.11 & 5.80 & 6.74 & 0.29 & 7.06 & 5.06 & 6.72 & 0.00 & 6.74 & 0.30 \\ \((2,15)\) & 6.79 & 7.03 & 3.53 & 7.60 & 11.93 & 8.64 & 27.25 & 6.79 & 0.00 & 6.80 & 0.15 \\ \((3,5)\) & 11.19 & 12.09 & 8.04 & 11.19 & 0.00 & 12.21 & 9.12 & 11.19 & 0.00 & 11.19 & 0.00 \\ \((3,10)\) & 10.29 & 10.75 & 4.47 & 11.04 & 7.29 & 11.84 & 15.06 & 10.29 & 0.00 & 10.32 & 0.29 \\ \((3,15)\) & 11.00 & 11.55 & 5.00 & 11.00 & 0.00 & 11.19 & 1.73 & 11.00 & 0.00 & 11.00 & 0.00 \\ \((4,5)\) & 13.89 & 14.15 & 1.87 & 13.89 & 0.00 & 14.72 & 5.98 & 13.89 & 0.00 & 13.89 & 0.00 \\ \((4,10)\) & 13.77 & 14.77 & 7.26 & 14.53 & 5.52 & 15.39 & 11.76 & 13.77 & 0.00 & 14.09 & 2.32 \\ \((4,15)\) & 14.37 & 15.76 & 9.67 & 14.97 & 4.18 & 15.79 & 9.88 & 14.37 & 0.00 & 14.46 & 0.63 \\ \hline Average & 10.53 & 11.23 & 6.90 & 10.92 & 4.27 & 11.71 & 12.46 & 10.53 & 0.00 & 10.58 & 0.41 \\ \hline \hline \end{tabular} Note: \(gap_{1}=\frac{obj-obj^{\prime}}{obj^{\prime}}\times 100\%\), where \(obj^{\prime}\) is the exact solution obtained by CPLEX.  
\end{table}
Table 1: Experimental results of small job instances with job processing time range from 5 to 10.  
\(LSA_{LPT}\), respectively. We thereby can conclude that (1) \(GA_{BM}\) outperforms \(ILSA_{LPT}\) in small-scale numerical experiments; and (2) the two proposed heuristic algorithms perform much better than the other three list algorithms.