4.1 Uniform Crossover

The Uniform crossover (UX) variation operator generates two children c1 and c2 by combining the genes of the parents solutions p1 and p2. In this operator the parents are considered in parallel, i.e. gene 1 of p1 taken on a par with gene 1 of p2. For every pair of genes of the parents, a random value with uniform distribution in the interval (0,1) is generated; a value less than or equal to 0.5 indicates that child c1 and c2 receive the gene from the parent p1 and p2 respectively, otherwise child c1 receives the gene from p2 and offspring c2 receives the gene from p1. During this process, there is the possibility of creating infeasible solutions, which is why heuristics are used to repair them. The performance of the UX operator has only been tested in a Grouping Genetic Algorithm for the Multiple Knapsack problem in 2008 [¹²]. In our implementation of UX for 1D-BPP, the genes with repeated items are eliminated from the children, and the missed items are reinserted with the FFD packing heuristic. Figure 3a depicts an example of the crossover process followed by UX.

Fig. 3 An example of each state-of-the-art crossover operator for a 1D-BPP instance: UX (Uniform Crossover), ESX (Exon shuffling Crossover), GPX (Greedy partition Crossover), and GLX (Gene-level Crossover) (Part 1) 

4.2 Exon Shuffling Crossover

The Exon Shuffling crossover (ESX), was proposed by Kolkman and Stemmer (2001) [¹⁴]. It has been often used to tackle 1D-BPP [⁹, ²⁵, ¹⁹]. This is an operator that generates a single child c from two parent solutions p1 and p2. The first step consists in joining the parents. Then their genes are ordered from best to worst with respect to their fullness. Finally, the genes are inherited to the child as long as none of the items of the respective gene exist previously in the child [²¹]. In our implementation of ESX for 1D-BPP, the missed items are reinserted with the FFD packing heuristic. Figure 3b depicts an example of the crossover process followed by ESX.

4.3 Greedy Partition Crossover

The Greedy Partition crossover (GPX), is also among the state-of-the-art grouping oriented crossover operators that have been used to tackle 1D-BPP in other Grouping Genetic Algorithm [²³]. This particular operator has two versions and this paper focuses on the group oriented one. Given two parent solutions p1 and p2, GPX generates two children c1 and c2 using a greedy heuristic. Here the first step is to order the genes of the parent solutions p1 and p2 from most to the least filled. Then, for each child, a vector of probabilities with uniform distribution of the size of the parent solution with more genes is generated. The probability defines from which parent the offspring will receive the gene; if the value generated is less than or equal to 0.5, it indicates that the child c1 receives the gene from p1; otherwise, it will receive the one from p2. The same process is employed to create the child c2. Like in the previous cases, in our implementation of GPX for 1D-BPP, the genes with repeated items are eliminated from the children and the missed items are reinserted with the FFD packing heuristic. Figure 4c depicts an example of the crossover process followed by GPX.

Fig. 4 An example of each state-of-the-art crossover operator for a 1D-BPP instance: UX (Uniform Crossover), ESX (Exon shuffling Crossover), GPX (Greedy partition Crossover), and GLX (Gene-level Crossover) (Part 2) 

4.4 Gene-Level Crossover

The operator is described in Section 3. This operator was proposed by Quiroz-Castellanos et al. [²⁰] for the GGA-CGT algorithm and has been used several times to solve the 1D-BPP due to its performance [¹⁵, ²², ²⁴]. Figure 4d depicts an example of the crossover process followed by GLX.

Since the above operators, are considered the best grouping crossover operators amongst the state-of-the-art, the next section comprises an analysis of these operators to determine which one enables the GGA-CGT to reach the best performance for 1D-BPP. The experiments cover: (1) the integration of the operators in the GGA-CGT algorithm; (2) the study of different crossover percentages; and, (3) the robustness analysis of the results of the best crossover operators inside the GGA-CGT algorithm.

5.1 State-of-the-Art Grouping Crossover Operators

The GGA-CGT employs the controlled reproduction technique proposed by Quiroz-Castellanos et al. [²⁰], for the state-of-the-art crossover operators we used the same strategy. Both, the UX operator and the GPX operator, generate a set C of nc children from nc parents. Like in the original GGA-CGT, the first nc/2 children are introduced to the population replacing the individuals in the set of random parents R. The other nc/2 children are introduced replacing individuals with repeated fitness and replacing the worst solutions. On the other hand, concerning the ESX operator, where nc/2 children are generated from nc parents (since only one child is generated for every two parents) the reintegration into the population is done in one way: the nc/2 children are introduced replacing the individuals in the set of random parents R.

Table 1, Table 2, Table 3, and Table 4 show the results obtained by the GGA-CGT with each of the state-of-the-art crossover operator (UX, ESX, GPX and GLX, respectively). For every class of instances, each table first shows the number of test cases (Inst.), followed by the number of optimal solutions found by the GGA-CGT with each crossover rate. The last row of each table shows the total number of optimal solutions obtained by each configuration. Moreover, Fig. 5 and Fig. 6 show the number of optimal solutions that are found and the average number of generations executed, when we explore ten different crossover rates (from 0% to 90%) over the four different crossover operators.

Fig. 5 Number of optimal solutions obtained by the GGA-CGT with each state-of-the-art crossover for different crossover rates 

Fig. 6 Average number of generations executed by the GGA-CGT with each state-of-the-art crossover for different crossover rates 

The results obtained from implementing the UX variation operator in the GGA-CGT algorithm, are shown in Table 1. As can be seen, the highest number of instances solved is 1592, with a crossover rate of 30%, however, with 90% of crossover rate, the performance of the GGA-CGT is affected since it only solves 1574 of the 1615 instances less than those solved with 0% crossover rate. In terms of the average number of generations, with a 90% of crossover rate, the algorithm iterates 100.22 generations per instance, as it is shown in Fig. 6.

The results obtained from implementing the ESX variation operator in the GGA-CGT algorithm are shown in Table 2. As can be seen, the highest number of the optimally solved instances is 1596, with a 50% crossover rate. Something interesting in this operator is that the optimally solved instances are between 1592 and 1596, and it is not affected with any crossover rate. Concerning to the average number of generations, the algorithm performs a maximum of 91.6 generations, corresponding to a crossover rate of 80%, as it is shown in Fig. 6.

The results obtained from implementing the GPX operator in the GGA-CGT algorithm, are shown in Table 3. The operator solves optimally at most 1595 instances with a 10% crossover rate. As the UX operator, the performance of the GGA-CGT is affected with a 90% crossover rate, since it only finds the optimal solution of 1576 instances less than those solved with 0% crossover rate. It is important to mention that it performs 99.56 generations on average with a 90% crossover rate, as it is shown in Fig. 6.

The same experiment was performed for the crossover operator of the GGA-CGT algorithm, GLX [²⁰]. The results are presented in the Table 4, which with a 20% crossover rate solves optimally 1602 instances out of 1615. The minimum number of optimal solutions is found with a crossover rate of 80%, being 1589, and it indeed benefits the algorithm, since without crossover, it finds only 1581 optimal solutions.

From Fig. 5 and Fig. 6 it can be observed that ESX and GLX outperformed the performance of the other two crossover operators, presenting a more stable behavior with different crossover rates.

After the execution of the GGA-CGT with the four crossover operators it was concluded that the ESX operator with a configuration of a 50% of crossover rate and the GLX operator with a 20% crossover rate, found a higher number of optimal solutions, which is why a series of experiments were performed, based on these results and they are described below.

5.2 Reinsertion of Children to the Population

Considering that the GLX algorithm generates two offspring for each pair of crossed parents, for the following experiments, the replacement criteria within the GGA-CGT algorithm were considered. Two versions of the GGA-CGT algorithm were implemented, where the GLX crossover operator generates only one child for each pair of parents.

For the implementation of the following experimentation, the following three cases were addressed: (1) when the ESX algorithm reinserts offspring into the population by replacing individuals with repeated fitness and, if there are still offspring without reinsertion, by replacing the worst solutions; (2) when the GLX algorithm reintegrates the created offspring into the population by replacing individuals within the set of random parents R; and, (3) as in the case of the ESX operator, when the GLX algorithm reinserts the offspring into the population by replacing individuals with repeated fitness and, if there are still offspring without reintegration, by replacing the worst solutions.

The case where ESX reinserts the children into the population by replacing individuals within the set of random parents R was presented in Table 2, Fig. 5 and Fig. 6. These results are used in Fig. 7 and Fig. 8 for the ESX-1 operator.

Fig. 7 Number of optimal solutions with different replacements. ESX-1 GLX-1: children replace the individuals in the set of the random parents. ESX-2, GLX-2: children replace the individuals with duplicated fitness and also the worst solutions 

Fig. 8 Average number of generations with different replacements. ESX-1 GLX-1: children replace the individuals in the set of the random parents. ESX-2, GLX-2: children replace the individuals with duplicated fitness and also the worst solutions 

The results obtained by the ESX variation operator, corresponding to the case when using reinsertion into the population by replacing the solutions within the repeated fitness group and worst solutions (ESX-2), are shown in Table 5. The GGA-CGT algorithm manages to solve optimally a maximum of 1593 instances with crossover rates of 30% and 40%. The lowest number of instances that it solves optimally is 1589 with a 10% crossover rate.

On the other hand, the results of the imple-mentation of the GLX operator with the reinsertion into the population by replacing the parents of the random group (GLX-1) are presented in Table 7. As it can be seen, the operator implemented within the algorithm finds a maximum number of 1599 optimal solutions with a 20% crossover rate, while the lowest number of optimal solutions is 1591 with a 80% crossover rate.

The last experiments, corresponding to the GLX with the reinsertion of the offspring in the population replacing the solutions with repeated fitness and worst fitness (GLX-2), are shown in Table 6. This operator allows the GGA-CGT to find a maximum of 1598 optimal solutions with crossover rates of 40%, 60% and 80%, while the minimum number of instances optimally solved is 1595 for 70% and 90% crossover rates, without taking into account the results when there is no crossover.

Based on the results discussed earlier in this section, with respect to the implementation of the two versions of the ESX operator, it is concluded that the ESX operator using a reinsertion of the children in the group of random parents (ESX-1), has a better performance, since it solves on average 1593.22 instances in an optimal way out of 1615. Moreover, the highest number of optimal solutions found by the ESX operator are solved with this reinsertion (1596). On the other hand, the other reinsertion, replacing parents with repeated and worst fitness solves an average of 1591.55 instances in an optimal way.

Regarding the results of the implementation of the GLX operator in the version that only creates one child for every two parents, when the reinsertion into the population is performed by replacing the solutions with repeated and worst fitness (GLX-2), it has better results, finding on average 1596.77 optimal solutions. In the case of the GLX operator with reinsertion to the population by replacing the set of random parents (GLX-1), an average of 1595.44 optimal solutions are found. Moreover, by means of this replacement the highest number of optimal solutions are found by the GLX operator (1599).

5.3 Uniform Crossover with Ordered Genes

After analyzing the different features involved in the crossover operators, we observe that the UX operator was the only one that did not sort the parent solutions in descending order before performing the crossover process. A final implementation of the UX operator was done, where the genes of the parents were first sorted in descending order of their filling, and then the usual crossover process of the operator was performed (UX-sorted). The results are shown in the Table 8. As we can see, the algorithm with a 10% crossover rate solves a total of 1596 instances in an optimal way, while with a 90% crossover rate it affects the performance of the algorithm as it only solves 1574 instances. In addition to the above, a comparison was made between the UX operator and UX-sorted. The results are shown in two tables, one presenting the maximum number of instances solved per class in the (Table 9) and the other with respect to the minimum number of instances solved per class in the (Table 10).

5.4 Robustness of Crossover Operators

To evaluate all of the potential of the best crossover operators, we performed a robustness test by executing three versions of the GGA-CGT thirty times with different seeds of random numbers. These last experiments arose from the comparison of the ESX and GLX operators.

For the GLX operator we used the version in which the children are reinserted in the population by replacing the random parents (GLX-1) with a 50% crossover rate, as well as the original version of the GLX included in the the GGA-CGT algorithm by Quiroz-Castellanos et al. [²⁰], that generates two children with a 20% crossover rate. On the other hand, for the ESX operator we also used the version in which the children are reinserted in the population by replacing the random parents (ESX-1) with a crossover rate of 50%. These operators were selected as they are within the group of operators that showed the higher effectiveness. The experiment consists of 30 executions of the GGA-CGT algorithm (48,450 runs) with each crossover operator.

The results of the experiments are shown in Table 11, Table 12 and Table 13. The second column (Inst.), corresponds to the total number of instances belonging to the class in the first column. The column “Optimal solutions in the 30 executions” includes the total number of optimal solutions found in the 30 executions.

The column “Average number of optimal” corresponds to the average number of instances optimally solved in each set. Finally, the column “%” corresponds to the percentage of optimal solutions found.

It is observed that the GGA-CGT algorithm with the GLX operator, which generates two children, found on average a higher number of optimal solutions, with a 98.99% percentage.

While the GLX and the ESX that generate one child found 98.8% and 98.67% of the optimal solutions, respectively. Furthermore, on average, the original GGA-CGT with the GLX operator that generates two children resolves in an optimal way an average of 1598.67 instances out of 1615. Regarding to the GLX and the ESX that generate one child, they found on average 1595.87 and 1593.53 optimal solutions, respectively.

5.5 Statistical Analysis of the Effectiveness of GGA-CGT with the Best Crossover Operators

A series of statistical tests were applied to the results of the 30 executions of the GGA-CGT algorithm with the ESX, GLX and GLX-V1 operators. The statistical test applied was the Wilcoxon Rank-sum test with a 95% confidence. These tests were performed on two result sets. First, the error of the results was calculated for each instance in each execution, which was determined as the relative difference defined as: (y−x)/x, where x corresponds to the optimal number of bins and y corresponds to the number of bins obtained.

For the first set of experiments, the average error of the 30 executions was obtained for each instance.

Next, the Wilcoxon test was applied to the average errors of the algorithm per class of instances with the three operators: ESX vs GLX-V1, ESX vs GLX and GLX-V1 vs GLX, obtaining 9 p-values (one for each class). The results are shown in the Table 15.

For the second set of tests, all instances were considered separately. A Wilcoxon test was performed for each instance with the results obtained from the algorithm with the different operators, i.e.: ESX vs GLX-V1, ESX vs GLX and GLX-V1 vs GLX. The Table 16 shows the results of the instances where a significant difference was observed in the results obtained with the different operators.

The Wilcoxon test, at the class level, shows no significant difference between the operators, which would indicate that the performance is the same among the three. However, the test at the instance level does show a difference, in some instances of the Hard28 and Gau 1 classes, between the ESX operator and the GLX-V1 and GLX operators, indicating that for these instances the ESX operator is the worst operator.

5.6 Difficult 1D-BPP Instances

Finally, with the objective to identify the features of the 1D-BPP instances that present the highest degree of difficulty for the four state-of-the-art grouping crossovers, Table 16 shows the set of instances for which the optimum was not found. These results were obtained from the set of experiments performed with the different crossover rates, selecting those instances that were not solved optimally in at least one of the configurations.

The first column includes the name of the sets of instances that include cases for which the optimal solutions could not been found for some of the configurations of the four crossover operators. The second column contains the name of the difficult instances, and the following columns include an X for each operator that fails to find the optimal solutions for each instance in one of its configurations. The instances belong to the classes: data set 1, data set 2, data set 3, triplets, uniform, hard28 and gau 1. The highest number of instances where the optimum is not found belong to the triplets class. With respect to the class data set 3 and uniform, only one instance is not optimally solved.

As it can be seen, the instances belonging to the hard28 class are the most complicated for the 4 operators. The operator with the lowest number of optimally solved instances is the ESX operator. Although the GLX operator optimally solves the highest number of instances, these instances tend to be more variable with different crossover percentages.

The UX operator, on the other hand, is the one that has the greatest diversity with respect to the number of instances not optimally resolved, being a total of 46. With respect to the instances belonging to the triplets class, the ESX operator is the only operator that has no problem in finding the optimum, besides being the only one to solve optimally all the instances of the Uniform class.

Although the GGA-CGT algorithm finds the optimal solution of most of the well-known benchmark instances within the state-of-the-art, it has been shown that it does not perform well with the 2800 new difficult instances, referred to as BPPvu_c, proposed by Carmona-Arroyo et al. [⁶]. The results that were obtained by the GGA-CGT demonstrated that for most of these instances, the operators included in the GGA-CGT do not appear to lead to better solutions.