# Workload Balancing on Identical Parallel Machines: Theoretical and Computational Analysis
## 5 Computational Analysis
### Results
#### 5.2.2 Dataset HGJ  
Table 3 lists the numerical results from solving the HGJ dataset using three models (MILP, MIQ1 and MIQ2). For this dataset, the gap of the performance between the three models is slightly smaller than that of the \(DM\) dataset. Accordingly, out of 560 instances, the percentage of instances solved optimally was 52% for MILP, 55% for MIQ1 and 51% for MIQ2. Altogether, 389 out of 560 instances were solved optimally, which is equivalent to 61%. Again, in the configuration whereby one model has some difficulties in finding a solution, the others can solve those instances more easily. For this dataset, the authors of [18] found 37 optimal solutions by solving the quadratic model by [PERSON] and 45 optimal solutions using a local search.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**M**} & \multirow{2}{*}{**N**} & \multicolumn{4}{c}{**Max Time**} & \multicolumn{4}{c}{**Min Time**} & \multicolumn{4}{c}{**Avg Time**} & \multicolumn{4}{c}{**Opt**} \\ \cline{3-14}  & & **MILP** & **MIQ1** & **MIQ2** & **MILP** & **MIQ1** & **MIQ2** & **MILP** & **MIQ1** & **MIQ2** & **MILP** & **MIQ1** & **MIQ2** & **All** \\ \hline \(15\) & 500 & 313.1 & 304.9 & 306.0 & 43.3 & 24.4 & 80.6 & 303.2 & 270.3 & 297.6 & 1 & 7 & 1 & 7 \\ \(15\) & 1000 & 323.9 & 304.1 & 303.0 & 305.5 & 108.0 & 300.1 & 313.0 & 229.6 & 300.6 & 0 & 22 & 0 & 22 \\ \(15\) & 2500 & 313.6 & 300.3 & 300.6 & 303.3 & 283.1 & 300.2 & 306.8 & 299.9 & 300.3 & 0 & 1 & 0 & 1 \\ \(15\) & 5000 & 315.5 & 300.6 & 301.0 & 300.4 & 72.8 & 300.5 & 305.5 & 285.5 & 300.6 & 0 & 6 & 0 & 6 \\ \(15\) & 10,000 & 301.6 & 301.1 & 302.0 & 300.5 & 28.6 & 300.7 & 300.9 & 193.0 & 301.1 & 0 & 30 & 0 & 30 \\ \hline \multicolumn{14}{c}{**Avg/**} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{228.5} & \multirow{2}{*}{110.5} & \multirow{2}{*}{190.4} & \multirow{2}{*}{500} & \multirow{2}{*}{1269} & \multirow{2}{*}{717} & \multirow{2}{*}{1370} \\ \multicolumn{14}{c}{**Total**} & & & & & & & & & & & & & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: \(Cont.\)  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**M**} & \multirow{2}{*}{**N**} & \multicolumn{4}{c}{**Max Time**} & \multicolumn{4}{c}{**Min Time**} & \multicolumn{4}{c}{**Avg Time**} & \multicolumn{4}{c}{**Opt**} \\ \cline{3-14}  & & **MILP** & **MIQ1** & **MIQ2** & **MILP** & **MIQ1** & **MIQ2** & **MILP** & **MIQ1** & **MIQ2** & **MILP** & **MIQ1** & **MIQ2** & **All** \\ \hline \(3\) & 10 & 0.2 & 0.1 & 0.2 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 20 & 20 & 20 & 20 \\ \(3\) & 12 & 0.3 & 0.2 & 0.2 & 0.1 & 0.1 & 0.1 & 0.1 & 0.2 & 20 & 20 & 20 & 20 \\ \(3\)  
Solving difficult instance of HGJ dataset (15, 20): we observed the same phenomenon in the DM dataset. In configuration (15, 20), the authors of [18] did not find an optimal solution; the quadratic model MIQ1 does not yield any optimal solution either. However, with a linear cut, MILP and MIQ2 found 20 optimal solutions out of 20 instances.