Experimental Study on the Sub-problem Generation Meth-

We now present experimental results to investigate how the three procedures to partition visits (LBU, RBK and SBK) and the three procedures to select work- force (BF, AF and WS) contribute to generating a final solution to the whole problem instance. The nine combinations are tested on the 42 problem instances and results are collected in terms of the solution quality and computation time. In the results presented here, LBU-BF denotes location based with uniform vis- its partition followed by best fitness workforce. A similar naming convention

is used for the other sub-problem generation procedures.

Figure 5.6 presents the summary of results comparing the nine sub-problem generation methods. From left to right, the figure shows the number of best solutions (#BestSolutions), average objective value (AverageObj) and average computational time (AverageTimes) in seconds. Each bar in each sub-figure shows the results obtained for all 42 instances when using one particular sub- problem generation method within RDCR.

In terms of number of best solutions, LBU-WS and SBK-WS achieve the highest number of best solutions (10 instances), followed by LBU-BF, RBK-BF and SBK-BF with 9 best solutions each. In terms of average objective value, eight of the methods gave very competitive results while only RBK-WS showed considerably lower performance.

In terms of average computational time, the figure seems to indicate that the LBU visits partitioning procedure combined with either BF or AF workforce selection are the fastest methods. The next fastest ones are the RBK visits par- titioning procedure combined with either BF or AF. The three methods using the WS visits partitioning method are the most time consuming. As mentioned

LBU-BF RBK-BF SBK-BF _{LBU-AF RBK-AF} SBK-AF _{LBU-WS RBK-WS} SBK-WS 0 5 10 9 9 9 6 6 6 10 7 10 # best solutions

LBU-BF RBK-BF SBK-BF _{LBU-AF RBK-AF} SBK-AF _{LBU-WS RBK-WS} SBK-WS 0 0.5 1 1.5 ×103 320.7 333.81 319.67 401.46 378.3 316.2 453.15 1,134.2 588.2 Objective values

LBU-BF RBK-BF SBK-BF _{LBU-AF RBK-AF} SBK-AF _{LBU-WS RBK-WS} SBK-WS 0 2 4 ×103 96.96 664.2 2,114 114.19 561.5 1,137.5 1,137.5 1,916.1 2,940.7 Computation time (s)

Figure 5.6:Overall results using the nine decomposition procedures within RDCR on the 42 HHC instances. The sub-figure on the left shows the number of best known solutions found with each procedure. The sub-figure in the middle shows the average objective value ob- tained with each procedure. The sub-figure on the right shows the average computational time in seconds when using each proced- ure.

Table 5.1:Friedman statistical test and mean ranks of objective value on 9 de- composition rules of RDCR. The lower mean rank presents better solution quality.

Friedman Test Mean Ranks

N 42 χ2 34.146 df 8 p <.001 Workforce Selection Task Partition LBU RBK SBK BF 4.44 4.50 4.31 AF 6.70 5.18 5.08 WS 4.98 5.83 3.98

before, we were expecting this to be the case as selecting all suitable workers increases the sub-problem size. However, we though that this workforce selec- tion method would result in better solutions but this was not the case as can be seen in the other sub-figures. We should note that there was a time limit set for solving each sub-problem of 30 seconds per visit.

We also conducted a statistical analysis using the non-parametric Fried- man’s ANOVA test to determine any statistically significant differences, in terms of solution quality and computation time, between the sub-problem generation methods. We used SPSS [63] and set the main significance level of the test at 0.05. Based on the results of this study we selected the LBU-BF method to be used within RDCR.

Table 5.1 reports the results of this test with the calculated statistic on the left and the mean ranks on the right. The results show significant differences

between the nine methods with χ2(8) = 34.146, p < .001. Therefore, we fol-

lowed this with pairwise comparisons to identify differences between groups. It showed that LBU-AF produced lower solution quality (higher objective value) than the other method. Overall, the decomposition method to be used with RDCR to find the best solution quality was SBK-WS because it had the lowest objective value mean rank.

In terms of computational time, the study identified three groups, with the methods giving lower computational time being LBU-BF and LBU-AF. Table

5.2 reports the results of this test with the calculated statistic on the left and the mean ranks on the right. Statistically significant differences were found among the nine methods. Furthermore, Table 5.3 summarises the pairwise compar- isons into three categories. The Positive column shows the number of other methods against which the method in the row spent more computational time with a statistically significant difference. Similarly, the Negative column shows the number of other methods against which the method in the row spent less computational time with statistically significant differences. Then, the Indif- ferent column shows the number of other methods against which the method did not reflect a significant difference on the computational time spent. Finally, the Category column classifies computational time of each decomposition rule into three groups: Faster, Middle, and Slower groups. In the first group are the faster methods: LBU-BF and LBU-AF. This group has two decomposition rules where they did not have positive pairwise differences. Note that more posit- ive differences means that the rule takes higher computational time than other rules. The second group are the rules with mixed results hence in the middle of the ranking: RBK-BF, SBK-BF, RBK-AF, SBK-AF and LBU-WS. The rules in this group are slower than the faster group but still have some negative dif- ference. Finally, there are two decomposition rules in the slower group which are RBK-WS and SBK-WS. The slower group does not have any negative dif- ferences. Hence, they require higher computational time to find a solution than other decomposition rules.

In summary, we presented a study on nine decomposition rules comparing their solution quality and computational efficiency. We applied statistical tests to find suitable decomposition rules considering on both factors. To get a high quality solution, the study showed indifference amongst the proposed rules ex- cept LBU-AF and SBK-AF. Nevertheless, the top three ranking were SBK-WS, LBU-BF and SBK-BF. The computational efficiency study presented decompos-

Table 5.2:Friedman statistical test and mean ranks of computational time on 9 decomposition rules of RDCR. The lower mean rank presents better solution quality.

Friedman Test Mean Ranks

N 42 χ2 161.118 df 8 p <.001 Workforce Selection Task Partition LBU RBK SBK BF 2.25 3.44 5.25 AF 3.33 3.94 5.64 WS 6.88 6.19 8.07

Table 5.3:Summation of differences in pairwise comparison between the 9 decomposition rules.

Number of pairwise differences

Category Decomposition rule Negative Indifferent Positive

LBU-BF 5 3 0 Faster LBU-AF 5 3 0 RBK-BF 3 4 1 Middle RBK-AF 2 5 1 SBK-BF 1 4 3 SBK-AF 1 4 3 LBU-WS 1 3 4 RBK-WS 0 4 4 Slower SBK-WS 0 2 6

ition rules in three groups. The rules which had higher computational effi- ciency were LBU-BF, RBK-BF and LBU-AF. Therefore, considering both evalu- ating factors, the selected decomposition rule for the next study was LBU-BF as its rank was one of the top three in both evaluation factors. Based on the results of this study we selected the LBU-BF method to be used within RDCR in a comparison with the other solution methods in the next section.

Number of best solutions Average objective value

GDCA GDCR RDCR Heuristic Human 0 10 20 30 0 15 27 2 0

GDCA GDCR RDCR Heuristic Human 0 2 4 ·104 17,213 338 320 446 48,089

Figure 5.7:The number of best known solutions (left sub-figure) and aver- age objective function (right sub-figure) obtained with the four al- gorithms and human solution (Human).