A greedy memetic algorithm for a multiobjective dynamic bin packing problem for storing cooling objects

Abstract

In this paper, a multiobjective dynamic bin packing problem for storing cooling objects is introduced along with a metaheuristic designed to work well in mixed-variable environments. The dynamic bin packing problem is based on cookie production at a bakery, where cookies arrive in batches at a cooling rack with limited capacity and are packed into boxes with three competing goals. The first is to minimize the number of boxes used. The second objective is to minimize the average initial heat of each box, and the third is to minimize the maximum time until the boxes can be moved to the storefront. The metaheuristic developed here incorporated greedy heuristics into an adaptive evolutionary framework with partial decomposition into clusters of solutions for the crossover operator. The new metaheuristic was applied to a variety benchmark bin packing problems and to a small and large version of the dynamic bin packing problem. It performed as well as other metaheuristics in the benchmark problems and produced more diverse solutions in the dynamic problems. It performed better overall in the small dynamic problem, but its performance could not be proven to be better or worse in the large dynamic problem.

Appendix

In (Fonseca et al. 2005), the authors advocate the use of the Kruskal-Wallis test to evaluate unary quality indicators with three or more independent samples. The unary indicators investigated here are the maximum spread indicators, the Pareto front absolute effiency, and the distance measures. Therefore, the Kruskal-Wallis test was applied to the samples to determine the presence of a signifcant difference before applying the student-t difference tests. The degrees of freedom for the student-t tests were found using the Satterthwaite approximation given in (39) (Satterthwaite 1946).

$$\begin{aligned} df = \frac{\Big (\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2}\Big )^2}{\frac{(S_1^2/n_1)^2}{n_1 - 1} + \frac{(S_2^2/n_2)^2}{n_2 - 1}} \end{aligned}$$

(39)

The statistical relevance of binary indicators are determined by two tests in this work. The Wilcoxon-rank sum test is performed first to determine if the samples in the comparison belong to the same distribution. If they are determined to be different, the student-t difference test is then performed to determine if the difference meets the conditions of the comparison operator, assuming similar standard deviations.

Applying this procedure to the binary coverage indicator, the Wilcoxon-rank sum test is used to prove either the null hypothesis and the alternative hypothesis shown in (40).

$$\begin{aligned} \begin{aligned} H_{wr,0}:&&I_C(A,B) = I_C(B,A)\\ H_{wr,1}:&&I_C(A,B) \ne I_C(B,A) \end{aligned} \end{aligned}$$

(40)

Then, if $H_{wr,1}$ is accepted, the student-t test is applied to determine if the difference between the samples corresponds to the null hypothesis, corresponding to the conditions for better performance, or to the alternative hypothesis shown in (41).

$$\begin{aligned} \begin{aligned} H_{st,0}:&&I_C(A,B) - I_C(B,A) \ge 1.0 - I_C(B,A) \\ H_{st,1}:&&I_C(A,B) - I_C(B,A) < 1.0 - I_C(B,A) \end{aligned} \end{aligned}$$

(41)

Applying this procedure to the binary-$\varepsilon$ indicator, the Wilcoxon-rank sum test is used to prove either the null hypothesis and the alternative hypothesis shown in (42).

$$\begin{aligned} \begin{aligned} H_{wr,0}:&&I_\varepsilon (A,B) = I_\varepsilon (B,A)\\ H_{wr,1}:&&I_\varepsilon (A,B) \ne I_\varepsilon (B,A) \end{aligned} \end{aligned}$$

(42)

Then, if $H_{wr,1}$ is accepted, the student-t test is applied to determine if the difference between the samples corresponds to the null hypothesis, corresponding to the conditions for better performance, or to the alternative hypothesis shown in (43).

$$\begin{aligned} \begin{aligned} H_{st,0}:&&I_\varepsilon (B,A) - I_\varepsilon (A,B) \ge I_\varepsilon (B,A) - 1.0 \\ H_{st,1}:&&I_\varepsilon (B,A) - I_\varepsilon (A,B) < I_\varepsilon (B,A) - 1.0 \end{aligned} \end{aligned}$$

(43)

Table 5 presents the statistical analysis of Fig. 5, using an overall Type-I error rate of 0.05% and Bonferroni’s Method (Abdi 2007) to evaluate if each comparison meets the necessary condition of $(I_C(A,B) = 1 \cap I_C(B, A) < 1)$. Table 5 shows that while the $I_C$ values for GAMMA-PC were proven to be significantly different than the values for MOMAD and MOEPSO, GAMMA-PC’s performance cannot be proven to be significantly better (or worse) than NSGA-II, MOMA, MOMAD, or MOEPSO by the binary coverage indicator.

Table 5 Statistical evaluation if GAMMA-PC performs better than NSGA-II, MOMA, MOMAD, or MOEPSO according to the $I_C(A, B)$ metric

Table 6 presents the statistical analysis of Fig. 6, using an overall Type I error rate of 0.05% and Bonferroni’s Method to evaluate if each binary comparison meets the condition of $(I_\varepsilon (A, B) \le 1 \cap I_\varepsilon (B, A) > 1)$. With this metric, GAMMA-PC is proven to perform better than NSGA-II, MOMAD, and MOEPSO but not MOMA.

Table 6 Statistical evaluation if GAMMA-PC performs better than NSGA-II, MOMA, MOMAD, or MOEPSO according to the $I_\varepsilon (A, B)$ metric

Table 7 Statistical evaluation if GAMMA-PC produces more diverse solutions than NSGA-II, MOMA, MOMAD, or MOEPSO by the maximum spread indicator for the static problem

Table 7 presents the statistical evaluation of the maximum spread indicator for the static problem. The Kruskal-Wallis test showed that there were statistically signficant differences in the maximum spread indicators for GAMMA-PC, NSGA-II, MOMA, MOMAD, and MOEPSO ($\chi ^2(2)= 83.3$, $p< 0.00001$). Based on this evidence, student-t tests were applied to the differences in the spread values between GAMMA-PC and the others. The alternative hypothesis was accepted for each test, indicating that GAMMA-PC produces more diverse solutions at a statistically significant level.

Table 8 Statistical evaluation if GAMMA-PC performs better than NSGA-II or MOMA by the Pareto front measures

Table 9 Toy problem: statistical evaluation if GAMMA-PC performs better than NSGA-II or MOMA according to the $I_C(A, B)$ metric

Table 10 Toy problem: statistical evaluation if GAMMA-PC performs better than NSGA-II or MOMA according to the $I_\varepsilon (A, B)$ metric

Table 8 presents the statistical evaluation of the box plots shown in Figs. 11 and 12. The Kruskal-Wallis test showed that there were statistically significant differences in the absolute efficiencies ($\chi ^2(2)= 7.89$, $p= 0.0194$), the average distances ($\chi ^2(2)= 9.62$, $p= 0.0081$), and the maximum distances ($\chi ^2(2)= 8.18$, $p= 0.0167$) for GAMMA-PC, NSGA-II, and MOMA. While the absolute efficiency of GAMMA-PC cannot be proven to be different, the difference between its distance and the other two algorithms’ is statistically significant.

Table 9 presents the statistical evaluation of the interpretation function for $I_C$. While it is proven that the values for GAMMA-PC belong to separate distributions than its counterparts, it cannot be proven that $(I_C($GAMMA-PC$,B) = 1 \cap I_C(B,$GAMMA-PC$) < 1)$ for either NSGA-II or MOMA. Therefore, its performance cannot be proven to be strictly better or worse than NSGA-II or MOMA by the binary coverage indicator.

Table 10 presents the statistical evaluation of Fig. 15. The samples of $I_\varepsilon$ values comparing GAMMA-PC to NSGA-II and MOMA were both proven to belong to different distributions than their counterparts and to meet the necessary condition of $(I_\varepsilon ($GAMMA-PC$, B) \le 1 \cap I_\varepsilon (B,$GAMMA-PC$) > 1)$. Comparing NSGA-II to MOMA showed that the difference in their $I_\varepsilon$ values was statistically significant, but neither met the necessary condition to show better performance.

Table 11 Statistical evaluation if GAMMA-PC produces more diverse solutions than NSGA-II or MOMA by the maximum spread indicator for the toy dynamic problem

Table 12 Full problem: statistical evaluation if GAMMA-PC performs better than NSGA-II or MOMA according to the $I_C(A, B)$ metric

Table 11 presents the statistical evaluation of the maximum spread indicator for the toy dynamic problem. The Kruskal-Wallis test showed that there were statistically significant differences in the maximum spread indicators for GAMMA-PC, NSGA-II, and MOMA ($\chi ^2(2)= 12.5$, $p= 0.0019$). Based on this evidence, student-t tests were applied to the differences between the spread values for GAMMA-PC and those for NSGA-II and MOMA respectively. Table 11 shows the results of these two tests, indicating that GAMMA-PC produces more diverse solutions than NSGA-II or MOMA.

Table 13 Full problem: statistical evaluation if GAMMA-PC performs better than NSGA-II or MOMA according to the $I_\varepsilon (A, B)$ metric

Table 14 Statistical evaluation if GAMMA-PC produces more diverse solutions than NSGA-II or MOMA by the maximum spread indicator for the full dynamic problem

Table 12 presents the statistical evaluation of Fig. 20. The distributions in all of the comparisons are shown to be significantly different, and none of the methods are proven to be better than the others.

Table 13 shows the statistical analysis of Fig. 21. It shows that the samples for GAMMA-PC are statistically different than its counterparts, but it does not support the hypothesis that GAMMA-PC performs better by this metric. MOMA is shown to have a lower average $I_\varepsilon$ value than NSGA-II in Fig. 21c, but it also cannot be proven to perform better.

Table 14 presents the statistical evaluation of the maximum spread indicator for the full dynamic problem. The Kruskal-Wallis test showed that there were statistically signficant differences in the maximum spread indicators for GAMMA-PC, NSGA-II, and MOMA ($\chi ^2(2)= 39.4$, $p< 0.00001$). Based on this evidence, student-t tests were applied to the differences in the spread values between GAMMA-PC and NSGA-II and between GAMMA-PC and MOMA. The alternative hypothesis was accepted for each test, indicating that GAMMA-PC produces more diverse solutions at a statistically significant level.