# Bi-Objective Dynamic Multiprocessor Open Shop Scheduling: An Exact Algorithm
## 4 Results and Discussion
### Characteristics of the Optimal Pareto Front  
The number of optimal Pareto front solutions (OPFs) obtained for each instance is an important characteristic for the studied problem. Among the 30 instances, it is found that the number of OPF ranges from 1 to 23. Figure 4 presents the frequency distribution of the number of OPFs for the 30 instances. It is evident that there is a high level of variability in that number. However, having a median of 5 indicates that the number of OPFs is generally few. Meanwhile, there is no statistical evidence that the number of OPFs is correlated with the size of the instance represented by the total number of operations which ranges from 27 to 38 in the randomly generated instances as demonstrated by the scatter plot in Figure 5.  
Figure 3: Computational times for the 30 instances.  
Another important characteristic is the wideness of the Pareto front which can be represented by the percentage difference between the maximum and the minimum values of both objective values for each instance. Dividing the wideness by the number of OPFs less than one provides an average indication of the spacing between solutions along the Pareto front for both objectives. This is referred to here as the spacing indicator. Therefore, for both objectives, the spacing indicators are evaluated as in Equations (15) and (16):  
\[\sigma_{C_{\max}} =\frac{\max\limits_{s\in\mathcal{P}}\left\{C_{\max}(s)\right\}- \min\limits_{s\in\mathcal{P}}\left\{C_{\max}(s)\right\}}{(|\mathcal{P}|-1) \times\min\limits_{s\in\mathcal{P}}\left\{C_{\max}(s)\right\}}\times 100 \tag{15}\] \[\sigma_{MWFT} =\frac{\max\limits_{s\in\mathcal{P}}\left\{MWFT(s)\right\}-\min \limits_{s\in\mathcal{P}}\left\{MWFT(s)\right\}}{(|\mathcal{P}|-1)\times\min \limits_{s\in\mathcal{P}}\left\{MWFT(s)\right\}}\times 100 \tag{16}\]  
It is found that the spacing for the makespan (\(\sigma_{C_{max}}\)) is mostly larger than the spacing for the MWFT (\(\sigma_{MWFT}\)) in 22 instances and the opposite happens only in five instances. Table 3 lists the descriptive statistics for both spacing indicators for 27 instances since only one optimal Pareto front solution is found in three instances, and, therefore, their corresponding values are excluded.  
Figure 4: A histogram showing the frequency distribution of the number of optimal Pareto front solutions.  
Figure 5: A scatter plot for the number of optimal Pareto front solutions versus the size of the instance.  
To the best of our knowledge, there is no standard reference to compare the spacing indicators of the OPFs of the DMOSP with. However, comparing the statistics for the two objective functions as summarized in Table 3 provides insights about the shape of the optimal Pareto front. The range, which is the difference between the maximum and the minimum observed values, is more than 8.5 times larger for \(\sigma_{\text{C}_{max}}\) than \(\sigma_{MWFT}\). Supported by the ratio of the standard deviation of \(\sigma_{\text{C}_{max}}\) to that of \(\sigma_{MWFT}\), it is evident that the Pareto front is more stretched in the direction of the makespan objective. For both spacing indicators, comparing the means with the median, first and third quartiles indicate that their distributions are skewed to the right. This means that the spacing indicators tend to have values that are relatively higher than the minimum reported values.