125 samples. It can be concluded that increasing fixture quantity decreases the average improvement percentage per cluster, the trend of which is variable.
The results mean that the intracluster order generated from Stage II is an improvement on the initial order generated in Stage I. The improvement results in a reconfiguration order on each fixture that minimises the pin manipulations required, thus decreasing the fixture reconfiguration time for operations on that fixture. The reduction in fixture reconfiguration time implicitly decreases the total makespan.
9.5 MILP Model Behaviour
126 Figure 9.5: Variables Growth Characteristics for Increasing Fixtures and Constant Parts
Figure 9.6 shows the increase in variables for the MILP model when the part quantity was increased for a constant number of fixtures. The fixture-part combinations displayed were also extracted from the 75 test runs detailed in Table A.24.
y = 949.94ln(x) + 1138 R² = 0.9744 y = 791.07ln(x) + 804.19
R² = 0.9833 y = 588.55ln(x) + 610.58
R² = 0.99
0 500 1000 1500 2000 2500 3000
0 1 2 3 4 5 6 7
Variables
Fixtures
Variables Growth Characteristics for Increasing Fixtures and Constant Parts
10 Parts 11 Parts 12 Parts
Log. (10 Parts) Log. (11 Parts) Log. (12 Parts)
127 Figure 9.6: Variables Growth Characteristics for Increasing Parts and Constant Fixtures
Figure 9.7 shows the number of nodes explored to solve each problem, in comparison to the number of variables that those problems consisted of.
Figure 9.7: Nodes Explored in relation to Variables
y = 0,9744x3,0091 R² = 0,9999 y = 1,2155x3,0264
R² = 0,9998 y = 1.3231x3.0369
R² = 0.9999
0 500 1000 1500 2000 2500 3000
0 2 4 6 8 10 12 14
Variables
Parts
Variables Growth Characteristics for Increasing Parts and Constant Fixtures
2 Fixtures 3 Fixtures 4 Fixtures
Power (2 Fixtures) Power (3 Fixtures) Power (4 Fixtures)
y = 33,231e0,0027x R² = 0,8844
0 50000 100000 150000 200000 250000 300000 350000
0 500 1000 1500 2000 2500 3000
Nodes
Variables
Variables vs. Nodes
Nodes Expon. (Nodes)
128 Figure 9.8 shows the solution time required to solve each problem, in comparison to the number of variables that those problems consisted of.
Figure 9.8: Solution Time required in relation to Variables
Figure 9.9 shows the solution time required to solve each problem, in comparison to the number of nodes explored to solve each of those problems.
Figure 9.9: Solution Time in comparison to Nodes Explored
9.5.4 Analysis
The results presented in Appendix A.3.1 reveal that the MILP solver converged to a solution for every case that was tested, i.e. a 100 % success rate for the problem range was achieved. Relative gaps of
< 0.1, as per the assigned solver settings, were observed for 27 cases (36 % of the test samples). The relative gaps resulted in absolute gaps that did not exceed 0.2 for most cases, apart from a value of 0.5 for one of the 12-part/2-fixture problems. The gaps, which were ≤ 0.5, were deemed acceptable due to the integer values that were used for the operation times, meaning that the final result would be the
y = 0,449e0,0026x R² = 0,9222
0 500 1000 1500 2000 2500 3000 3500
0 500 1000 1500 2000 2500 3000
Solution Time (s)
Variables
Variables vs. Solution Time
Solution Time Expon. (Solution Time)
y = 0,0095x + 5,6625 R² = 0,9864
0 500 1000 1500 2000 2500 3000 3500
0 50000 100000 150000 200000 250000 300000 350000
Solution Time (s)
Nodes
Nodes vs. Solution Time
Solution Time Linear (Solution Time)
129 same as that achieved if the relative gap was zero. The relaxed settings allowed for reduced solution times to be achieved when possible, due to the reduction in nodes explored.
An average of 1.867 feasible solutions were found for the problems tested; no more than 5 feasible solutions were found for any problem, while 1 feasible solution was the most frequent outcome. Rough testing revealed that using the ‘most fractional’ branch rule produced the optimal solution by exploring fewer nodes than the ‘max fun’ option (as mentioned in Section 6.9.3); this would explain why minimal feasible solutions were found, with the benefit of finding the optimal solution in reduced time.
Constraint violations were observed in three cases; the order of magnitude of these violations were no greater than 10-15, which is negligible in comparison to the parameters of the problem (order of magnitude 100). Therefore, these results were deemed satisfactory and the solutions regarded as optimal.
Figure 9.5 shows the trend lines for each case (number of parts) that was investigated. The coefficients of determination (R2) confirmed that the growth characteristic of variables when fixtures were increased was logarithmic. The logarithmic growth is shown to increase for increasing part quantities; the sharpest increase in the number of variables is observed for 12 parts, whereas the 10-part plot is shown to be almost linear. The results reveal that increasing the fixture quantity for a constant number of parts causes the variable size to grow, albeit very slowly.
Figure 9.6 shows the trend lines for each case (number of fixtures) that was investigated. The coefficients of determination (R2) confirmed that the growth characteristic of variables when parts were increased was polynomial, roughly to the power 3. The rate of polynomial growth is shown to increase slightly for increasing fixture quantities. The equations generated from the graphs confirm the observation of sharper increases for greater fixture quantities via the slightly increasing values of the exponents (3.0091, 3.0264 and then 3.0369). The results reveal that increasing the part quantity for a constant number of fixtures causes the variable size to grow very quickly. It is suggested that it is this characteristic that prevented the MATLAB® MILP solver from solving problems consisting of more than 12 parts.
Figure 9.7 shows a large distribution of the data points. An outlier is observed for one of the 12-part/5-fixture problems; the solution to this problem required the exploration of a far greater number of nodes than its counterparts. The general distribution of the data was notably varied, even when disregarding the outlier. The trend line was constructed with a coefficient of determination (R2) of 0.8844, which is not ideal, but deemed sufficiently high to recognise the general trend of the observation. The trend observed was that of exponential growth, whereby the nodes explored to find the solution for larger problems (in term of number of variables) grew at a sharply increasing rate; this is only a general trend, as the variation along the trend line is significant. Therefore, there is no definite correlation between the number of variables of a problem and the nodes required to be explored by the branch and bound algorithm to solve it; however, a broad trend of exponential increase is observed, such that solving larger-sized problem would be expected to take longer to solve (with an extent of uncertainty).
Figure 9.8 displays similar characteristics to Figure 9.7. An exponential trend was observed, with a higher coefficient of determination than that of Figure 9.7. The outlier from Figure 9.7 is also detected
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.