## 4 Results and Discussion
### Data Set 2 Benchmarking Instances
#### 4.2.1 Instances with Ratio Less Than 0.25  
Table 4 shows the performance comparisons against those by the MILP model and LAHC heuristic of [PERSON] and [PERSON] [18] and the DE of [PERSON], [PERSON] and [PERSON] [11].  
The two-phase heuristic matches 17 out of 19 optimal solutions obtained by the MILP model. EX_940 is superior to its LAHC and DE counterparts.  
For five instances (EX_710, EX_720, EX_730, EX_740, EX_741), the two-phase heuristic obtained superior solutions to those of LAHC. The generated solutions for EX_710, EX_720, EX_730, and EX_740 are shown in Figures 6-9 respectively. The superior solutions obtained for EX_730 and EX_740 is attributable to _job pre-emption_.  
In EX_730, Job 7 is transported from L/U to M1 by V1 and stays in M1 until it has completed all processing. Upon arriving at M4 at 8 time units, its first and second operations (J7_O1, J7_O2) are processed consecutively until 48 time units, then _pre-empted_, unloaded, and _placed in the buffer_. This pre-emption is caused by Job 3, which has arrived from M4 at 44 time units and is _placed in the buffer_ and only starts processing its second operation from 48 up until 60 time units. Job 7 (J7_O3) is then _reloaded immediately_ for processing from 60 up until 82 time units.  
\begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c c c c c c c c c} \multicolumn{13}{c}{Characteristics} & \multicolumn{1}{c}{2-Phase} & \multicolumn{1}{c}{**MLP**} & \multicolumn{1}{c}{**LAHC (1000)**} & \multicolumn{1}{c}{**PDE-1**} & \multicolumn{1}{c}{**PDE-2**} \\ \hline
**Instance** & **M** & **J** & **O** & **A** & **Nodes**  
In EX_740, Job 4 is transported from L/U to M2 by V1 and stays in M2 until it has completed all processing. Upon arriving at M4 at 18 time units, its first operation (J4_O1) is processed consecutively until 46 time units, then _pre-empted_, unloaded, and _placed in the buffer_. This pre-emption is caused by Job 8 and Job 1. Job 8 arrives from L/U at 46 units and only starts processing its second operation immediately until 82 time units and is transported away to M4 for further processing. Following this, the second operation of Job 1 (J1_O2), which has arrived at M2 at 65 time units and was _placed in the buffer_, starts processing from 82 to 92 time units. Finally, Job 4 (J4_O2) is then _reloaded immediately_ for processing from 92 to 104 time units.  
The two-phase heuristic is more robust than LAHC for most instances, as evidenced by having percentage makespan standard deviation values that are mostly zero in value or are lower than its LAHC counterpart, except for in two instances.  
As mentioned for data set 1, comparing both the MILP model timings and that of the two-phase heuristic for instances hitting optimality, it is observed that the latter is much faster than the former. This supports the usage of the decomposition approach as opposed to solving the problem monolithically. Furthermore, this is highlighted by the two-phase heuristic locating the optimal solution of EX_241, EX_441, EX_930 while LAHC is unable to do so. Compared to the MILP model, EX_441 is able to achieve this at a miniscule timing (63.43 s versus 12,727 s).  
Figure 8: Gantt chart for improved solution of EX_730 instance from [PERSON], [PERSON] and [PERSON] [11].  
Figure 6: Gantt chart for improved solution of EX_710 instance from [PERSON], [PERSON] and [PERSON] [11].  
Figure 7: Gantt chart for improved solution of EX_710 instance from [PERSON], [PERSON] and [PERSON] [11].  
Figure 9: Gantt chart for improved solution of EX_740 instance from [PERSON], [PERSON] and [PERSON] [11].