130 in Figure 9.8, with the other data points being similarly scattered as in that previous figure. It can be concluded that the general trend of solution time increase in comparison to problem size (in terms of number of variables) is an exponential increase; however, the degree of uncertainty detected in Figure 9.8 also applies here.
The relationship between nodes explored and solution time was investigated, due to the similarities observed between nodes explored and solution time required when compared to the number variables.
Figure 9.9 shows that a linear relationship exists between the solution time and nodes explored. The trend line is shown to agree with the previously observed outlier as well. It can be concluded that the solution time relies on the number of nodes explored to find the solution. The number of nodes required varies greatly with each problem, which means that accurately predicting the solution time for a problem is impracticable. However, the trends observed in Figure 9.7 and Figure 9.8 do suggest an exponential increase in nodes explored, and thus solution time; this agrees with the expected behaviour of the branch and bound algorithm solution time, which is said to increase exponentially with the size of a problem [113].
9.5.5 Conclusion
The optimal solution was found for the problem range tested, verified by the convergence of the lower and upper bounds of the branch and bound solver (see Appendix A.4.1). Non-idealities in some of the results were discussed and it was deduced that the solutions were optimal nonetheless, given the parameters used.
The number of variables was found to increase logarithmically for increasing fixtures. The number of variables was found to increase polynomially for increasing parts, roughly to the power 3. The increase in both nodes and solution time in comparison to number of variables was found to be exponential. The relationship between nodes and solution time was verified to be directly proportional.
The sharp increase in variables for increasing fixtures, together with the exponential increase in nodes and solution time for increasing variables, insinuates why the solver was unable to solve problems with part quantity greater than 12. The inability was likely due to the high computational expense required to solve such problems to optimality.
131 times as per the seeds used for the MILP problems. The resultant total idle times were compared to inspect the degree of sub-optimality of the S3H in comparison to the optimal solutions. The solution times to obtain the solutions were also compared thereafter.
Test range: Selected problems from Section 9.5
Repetitions per test: 3
Criterion: Total idle time, Solution time 9.6.3 Results
Figure 9.10 displays the discrepancies that existed between the total idle times generated by the Stage III Heuristic in comparison to the optimal solutions generated by the MILP model for the same problems.
The specific total idle times are shown in the table incorporated below the graph. The results were generated from Appendix A.5.
Figure 9.10: Heuristic Solution discrepancies in comparison to Optimal Solutions
The individual percentage discrepancies were calculated and synthesised. The results are summarised in Table 9.4.
Table 9.4: Discrepancy results synthesis
Average (%) Median (%) Minimum (%) Maximum (%)
40.17 22.85 0 150
Figure 9.11 shows the discrepancy percentages between heuristic and optimal solutions when the problems were arranged according to increasing fixture quantity.
1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930313233343536 S3H solution 313883901161862993179714146911811473172616151815252319151814141211108312 MILP solution 303883391161502854109713106654741069122416831693252319121410741057424357
0 50 100 150 200 250 300
Total Idle Time (s)
Heuristic Solutions vs. Optimal Solutions
132 Figure 9.11: Discrepancy Increase in terms of Fixture Increase
Figure 9.12 shows the discrepancy percentages between heuristic and optimal solutions when the problems were arranged according to increasing part quantity.
Figure 9.12: Discrepancy in terms of Part Increase
Figure 9.13 shows the comparison between the Stage III Heuristic solution times and those of the MILP model for the same test problems. The test problems are arranged in order of increasing variables (and thus, complexity) of the MILP model problem formulation.
0 20 40 60 80 100 120 140 160
0 3 6 9 12 15 18 21 24 27 30 33 36
Discrepancy (%)
Problems
Discrepancy Percentages in terms of Fixture Increase
0 20 40 60 80 100 120 140 160
0 3 6 9 12 15 18 21 24 27 30 33 36
Discrepancy (%)
Problems
Discrepancy Percentages in terms of Part Increase
133 Figure 9.13: Heuristic Solution Times in comparison to MILP Solution Times
9.6.4 Analysis
The average heuristic solution was just above 40 % greater than that of the optimal solution. The median, however, was much lower at 22.85 %; this indicates that the majority of solutions were closer to the optimal solution than within 40 %. The maximum discrepancy was found to be 150 % greater than the optimal solution, generated for one of the 12-part/6-fixture problems (which were the most complex in terms of variables for the MILP model). The heuristic produced solutions that were identical to the optimal solution in nine instances, which made up 25 % of the sample set. The solutions only matched for 2-fixture problems. The observation can be explained by the relative lack of feasible solutions available to the solver in these cases, due to the combination of the following factors: the same fixture must not be scheduled in consecutive time periods, and; the intracluster order must be upheld.
The solutions do not always coincide for 2-fixture problems, as shown by six of the 2-fixture problems tested, i.e. 40 % of such cases. However, it should be noted that the 2-fixture problems would have skewed the data in favour of the heuristic.
Figure 9.11 shows the behaviour of the discrepancy percentages when the problems were arranged according to increasing fixture quantity. Given the phenomenon that occurred for many 2-fixture problems, it was expected that the discrepancies may increase continuously; however, the graph revealed that the discrepancies fluctuated greatly, and no trend line could match the data points with an agreeable coefficient of determination.
Figure 9.12 shows the behaviour of the discrepancy percentages when the problems were arranged according to increasing part quantity. The discrepancies showed high fluctuation as for Figure 9.11, and an agreeable coefficient of determination was similarly elusive.
Figure 9.13 shows the superiority of the S3H in terms of solution times for the same problems. The solution time discrepancies remain marginal until the complexity of the MILP problem reaches 1000 variables, which is where a significant divergence emerges. The S3H solution time continues to remain
0 50 100 150 200 250 300 350 400 450
64 64 216 276 276 512 736 736 945 1000 1000 1540 1728 1728 2256 2520 2520 2784
Solution Time (s)
Variables for MILP Problem
Heuristic Solution Times vs. MILP Solution Times
S3H time MILP time
134 under 0.2 seconds, while the MILP solution reaches solution times close to 400 seconds for the sample set. As stated in Section 9.5, the MILP model is unable to reliably solve problems with greater than 12 fixtures. The S3H, however, was able to solve a 12000-part/600-fixture problem in 1.6135 seconds.
9.6.5 Conclusion
The heuristic performed reasonably well in comparison to the MILP model, with discrepancies in total idle time averaging 40.17 % for the sample set. The discrepancies showed no definite trend in relation to problem complexity, apart from the most complex problems producing some of the highest discrepancies. This was expected due to the greater availability of feasible solutions that the B&B solver can explore; as opposed to the greedy approach of the S3H, which is unable to backtrack to find better solutions.
Solution times for the heuristic were almost negligible; every problem was solved within 0.2 seconds.
It can be concluded that the heuristic provides good, feasible (near-optimal) solutions with substantial savings in solution time.