Results for Memory Embedded Artificial Bee Colony (MEABC) Algorithm

The reason for this condition is probably linked to characteristics of item-sets. As defined above, two different distributed grouped item-sets consist of integer and decimal sized items. Obtained results show that the BFD algorithm cannot show sufficient performance with decimal sized items. The possible reason for this condition is that, if decimal sized items are sorted in a descended trend, items may not be allocated in bins because the same fractioned numbers possibly cannot be assembled easily. On the other hand, different fractioned numbers might be assembled with higher possibility in comparison with the mentioned arrangement. As a next and last step, the results of the proposed memory embedded artificial bee colony (MEABC) algorithm are illustrated in table 6.A.

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Table 6.A: Results of Memory Embedded ABC Algorithm for Itemset-1 Group-1

Fig. 30.A: Average Number of Bins Graph for Table 6.A Fig. 30.B: Average Wastage Graph for Table 6.A

According to results in table 6.A, the MEABC algorithm obtains the exact solution like the classic ABC algorithm because of the smallness of the item-set size. This condition changes in next experiments with an increase in wastage level. In Fig. 30.A and 30.B; graphs obtain the best solution of the process at near iteration 1000 and then they are stabilized with a worse solution. Conversely, other sub-groups of itemset-1 (Appendix C page 129-132) reach their stabile condition earlier than sub- group 1 however; sub-group 17 shows sharp change at the end of experiment run. This means that, proposed method may be executed for more iteration.

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Fig. 31.A: Average Number of Bins for Table 6.B Fig. 31.B: Average Wastage for Table 6.B

Like other results, in Fig. 31.A and Fig. 31.B; both graphs tend to achieve stability after an amount of iterations. The straight line of the graphs could be considered as the average of the results if it is achieved in a short time. According to the graphs above, the best solution of the experiment is obtained at near iteration 7000 and then the graphs return to the straight line representing a worse solution after a number of iterations.

Table 6.C: Results of Memory Embedded ABC Algorithm for Itemset-3, Itemset-4, and Itemset-8

Fig. 32.A: Average No. of Bins Graph for IS3 in Table 6C Fig. 32.B: Average Wastage Graph for IS3 in Table 6C The results of the MEABC algorithm for itemset-3 are the same as the results of the ABC algorithm and the graphs show the same characteristics. It means that the memory system does not affect the ABC algorithm search results. Besides, MEABC results for sub-group 13 (Appendix B, page 93, Table 11) of itemset-3 obtains better solution than classic ABC algorithm as an exception. Itemset-3, itemset-4 and itemset-8 consist of large number of items and run times for them are much longer

BIN WASTAGE (%) TIME (sec) MEABC IS-3 Gr-3 100-7500-100 207 0.0280 5.7224e+03 MEABC IS-4 Gr-1 100-7500-100 411 0.0306 1.4848e+04 MEABC IS-8 Gr-8 100-7500-100 175 0.0473 6.1104e+03

71 than others and because of this circumstance each experiment is run once. In this case, exceptional results are obtained just for aforementioned item-sets because of randomness.

Fig. 33.A: Average No. of Bins Graph for IS4 in Table 6C Fig. 33.B: Average Wastage Graph for IS4 in Table 6C In Fig. 33.A and 33.B; both graphs also show the same performance as the classic ABC algorithm search results and cannot differentiate the obtained results.

Table 6.D: Results of Memory Embedded ABC Algorithm for Itemset-5 Group-2

Fig. 34.A: Average Number of Bins Graph for Table 6.D Fig. 34.B: Average Wastage Graph for Table 6.D

The graph of average results for itemset-5 with the MEABC algorithm search is slightly different from the graph of the ABC algorithm search due to the increased number of plunges in the graph. On the other hand, graphs in Fig. 34.A and 34.B can be considered as unstable in comparison with graphs of other sub-groups of itemset-5 (Appendix C page 144-147).

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Table 6.E: Results of Memory Embedded ABC Algorithm for Itemset-6 Group-2

Fig. 35.A: Average Number of Bins Graph for Table 6.E Fig. 35.B: Average Wastage Graph for Table 6.E

In contradistinction to all other graphs, in Fig. 35.A and 35.B; graphs show interesting characteristics with two different straight lines. The trend of the second line observed after iteration 5000 represents a worse solution than the previous worse solution in comparison with the obtained best solution and cannot be fixed until the end of the process. In spite of this, the global optimum is the same as the solution obtained by ABC algorithm search.

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Fig. 36.A: Average Number of Bins Graph for Table 6.F Fig. 36.B: Average Wastage Graph for Table 6.F

The graphs of classic ABC algorithm search and MEABC algorithm search obtain the same optimum solution; however, the MEABC algorithm search reduces the number of instant changes and stabilizes the graph in a shorter time.

Fig. 37.A: Average No. of Bins Graph for IS8 in Table 6C Fig. 37.B: Average Wastage Graph for IS8 in Table 6.C The MEABC algorithm search obtains a more proper graph than the classic ABC algorithm especially for wide sized item-sets like those shown in Fig. 37.A and Fig. 37.B.

After all results are presented separately, table 7.A and 7.B provide an overview about algorithms using itemset-4 and itemset-8 due to their huge set size because the difference between algorithms cannot be easily observed when algorithms use small sized itemsets. Numbers at the left side of tables represent which sub-group of item-set is used.

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Table 7.A: Summary Result Table for Itemset-4

Table 7.B: Summary Result Table for Itemset-8

According to the results of the MEABC algorithm, each approach obtains almost the same solution for all item-sets. Classic ABC algorithm generally beats the approached MEABC algorithm. On the other hand, the MEABC algorithm obtains better results than the classic ABC algorithm as illustrated in Table 11 (Appendix B page 93) for experiment group DS13, in table 12 (Appendix B page 96) for experiment group DS17, and in Table 16 (Appendix B page 124) for experiment group DS19 probably because of experimental run randomness.

In fact, throughout this study, different subsidiary approach techniques with various parameter values were applied to improve the classic ABC algorithm; however, in the 1-D bin packing algorithm, the proposed approach cannot produce significant results to be considered as an improvement. Possible reasons for this case can be considered as follows:

 It is a fact that the ABC algorithm could be more effective for real number based functions, so the bin packing problem is a branch of integer programming and the main drawback of this method is that there are lots of solutions in the feasible area between two integers and this

75 method cannot benefit these solutions. In the 1-D bin packing problem, algorithms from literature may not find the actual solution.

 Both algorithms restrict the search by using the counter operator in classic ABC algorithm and tabu period in the proposed MEABC algorithm. These operators may prevent a further search in the feasible area.

 The classic and proposed algorithm might require a higher number of iterations to reach the optimum solution; however, in this case the application of these algorithms can reach a high cost. At least the mentioned algorithm run times should be reasonably short to be considered as a candidate run time because traditional algorithms solve the problem in a very short time.  The classic ABC algorithm also has its own control parameter unlike in Pham et al.’s bees

algorithm. In this case, it is unavoidable that both classic ABC algorithm and proposed MEABC algorithm obtain the same optimum solutions. Although the MEABC algorithm proposes a better memory system, the classic ABC algorithm also has a kind of immature short term memory. According to all the results, it seems that the classic ABC algorithm developed by Karaboga does not require a memory system at least for integer programming.

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