We conducted an experiment to study the GDCA performance. The flow of the study is depicted in Figure 4.1. The figure outlines the three parts of the experi- mental design. First, on the left-hand side of the figure, the permutation study refers to solving the sub-problems in different orders given by all the different
Figure 4.1:Outline of the experimental study in three parts: permutation study, observation step and strategies study.
permutations of the geographical regions. However, trying all permutations is practical only in small problems. Therefore, finding an effective ordering pattern is the second part of the experiment, observation step in the figure. This second part solves each sub-problem using all available workforce, i.e. ignoring whether some workers were assigned in previous sub-problems. The third part analysed the results from the observation step in order to define some strategies to tackle the sub-problems. Based on this strategies study, some solv- ing strategies were conceived. Listed in the figure are these ordering strategies: Asc-task, Desc-task, Asc-w&u, etc. More details about these ordering strategies are provided when describing the Observation step below. Finally, the solu- tions produced with the different ordering strategies are compared to the solu- tions produced by the permutation study to evaluate the performance of these ordering strategies.
with the number of geographical regions, we performed the permutation study
using only the instances with|A| = 3 and|A| =4 geographical regions where
the number of permutation is managable. Figure 4.2 shows the relative gap obtained for the small instances that have 3 regions. Each sub-figure shows the results for one instance when solved using the different permutation orders of the 3 regions. Each bar shows the relative gap between the solution by the decomposition method and the overall optimal solution. The figure shows that the quality of the obtained solutions for the different permutations fluctuates considerably. Closer inspection reveals that in these instances the geographical regions are very close to each other and sometimes there is an overlap between them. The result also reveals that some permutations clearly give better results. For example, permutation “1-2-3” for instance A-04, permutations “1-2-3” and “2-1-3” for instance A-05 and permutation “1-3-2” for instance A-07.
Figure 4.3 shows the relative gap obtained for the small instances that have 4 regions. Each sub-figure shows the result for one instance when solved us- ing the permutation orders of the 4 regions. Each bar shows the relative gap between the solution by the decomposition method and the overall optimal solution. Results in Figure 4.3 indicate that some solutions obtained with the decomposition approach using some permutations have a considerable gap in quality compared to the overall optimal solution. The figure also shows that
1-2-3 1-3-2 2-1-3 2-3-1 3-1-2 3-2-1 0 200 400 600 Relative gap (%) A-04 1-2-3 1-3-2 2-1-3 2-3-1 3-1-2 3-2-1 20 30 40 A-05 1-2-3 1-3-2 2-1-3 2-3-1 3-1-2 3-2-1 10 20 30 40 A-07
Figure 4.2:Relative gap obtained from solving the 3 instances (A-04, A-05 and A-07) with|A| = 3 using the different permutation orders. Each graph shows results for one instance. The bars represent the re- lative gap between the solution obtained with the decomposition method and the overall optimal solution.
20 40 60 80 100 Relative gap (%) A-02 0 5 10 Relative gap (%) B-02 1-2-3-4 1-2-4-3 1-3-2-4 1-3-4-2 1-4-2-3 1-4-3-2 2-1-3-4 2-1-4-3 2-3-1-4 2-3-4-1 2-4-1-3 2-4-3-1 3-1-2-4 3-1-4-2 3-2-1-4 3-2-4-1 3-4-1-2 3-4-2-1 4-1-2-3 4-1-3-2 4-2-1-3 4-2-3-1 4-3-1-2 4-3-2-1 5 10 15 20 Relative gap (%) B-04
Figure 4.3:Relative gap obtained from solving the 3 instances (A-02, B-02 and B-04) with |A| = 4 using the different permutation orders. Each graph shows results for one instance. The bars represent the re- lative gap between the solution obtained with the decomposition method and the overall optimal solution.
some permutations clearly give better results than others. For example, per- mutations “2-4-1-3”, “2-4-3-1” and “3-2-4-1” for instance A-02, permutations “1-2-3-4”, “1-2-4-3”, “2-1-3-4”, “2-1-4-3” and “2-3-1-4” for instance B-02 and permutations “4-3-1-2” and “4-3-2-1” for instance B-04.
The conclusion from this permutation study is that the order in which the sub-problems are solved matters differently according to the problem instance. More importantly, the results confirm our assumption that some particular per- mutations could produce a very good result in the decomposition approach. Hence, the next part of the study is to find a good solving order.
Observation step. Here we solve each of the sub-problems using all avail- able workers and collect the following values from the obtained solutions: num- ber of visits in the sub-problem (# visit), minimum number of workers required in the solution (# min worker), number of unassigned visits in the solution (# unassigned visit) and the ratio of visits to worker in the solution (visit/worker ratio). Then, we defined six ordering strategies as follows. Increasing number
Table 4.1:GDCA solution gap to the optimal solution of 14 smaller instances by six ordering strategies.
Instance Asc-Task Desc-Task Asc-w&u Desc-w&u Asc-Ratio Desc-Ratio
A-01 24.05 62.04 24.05 62.04 24.05 62.04 A-02 41.87 81.94 107.40 45.07 46.79 48.60 A-03 151.59 255.46 151.59 255.46 204.73 154.05 A-04 8.66 117.57 8.66 117.57 8.66 117.57 A-05 14.28 46.20 14.28 46.20 14.28 46.20 A-06 0.00 5.48 0.00 5.48 0.00 5.48 A-07 41.34 29.43 13.57 29.43 41.34 29.43 B-01 17.51 5.07 14.07 14.66 17.91 5.15 B-02 10.46 7.89 10.46 0.00 5.30 8.95 B-03 30.70 19.77 85.78 56.55 83.99 21.26 B-04 10.20 6.70 10.20 6.70 19.11 6.76 B-05 207.22 160.06 126.53 271.78 158.14 130.43 B-06 61.78 55.28 54.99 36.46 151.54 114.09 B-07 140.24 126.60 104.48 126.86 244.32 182.41 Average 54.28 69.96 51.86 76.73 72.87 66.60
Boldtext refers to the best solution.
of visits in the sub-problem (Asc-task); decreasing number of visits in the sub- problem (Desc-task); increasing sum of minimum workers required and unas- signed visits (Asc-w&u); decreasing sum of minimum workers required and unassigned visits (Desc-w&u); increasing ratio of visits to worker (Asc-ratio) and decreasing ratio of visits to worker (Desc-ratio).
Strategies study. The GDCA approach is again executed using the 6 order- ing strategies listed above to tackle the sub-problems in each problem instance. The results are presented in Table 4.1 which shows the relative gap for the 14 small instances in the A and B groups. Note that each value represents the relative gap obtained with each strategy.
Table 4.1 presents GDCA solution relative gap to the optimal solution of the 14 smaller instances when applying six different ordering strategies. These 14 smaller instances are the HHC instance sets A and B where the optimal solu- tion can be found by solving the problem as a whole. The results in the table are grouped by instance sets and the last row presents average relative gaps to the optimal solution. Overall, the decomposition technique with ordering strategies gives solutions with relative gaps up to 270% with 65% on average. The results show that some of the ordering strategies are more likely to produce
Table 4.2:Relative gap (%) of best permutation VS. best strategy. Set B.Perm B.Strt Set B.Perm B.Strt
A-04 3.53 8.66 A-02 20.41 41.87
A-05 14.28 14.28 B-02 0 0
A-07 9.91 13.57 B-04 4.07 6.70
better solutions than others. The best performing ordering strategy is Asc-w&u that gives 8 best solutions considering all 14 small instances. The average gap for the ordering strategies Asc-task, Desc-task, Asc-w&u, Desc-w&u, Asc-ratio and Desc-ratio are 54.28%, 69.96%, 51.86%, 76.73%, 72.87% and 66.60% respect- ively. On this occasion, we considered solutions with gap more than 100% as poor solutions. Thus, the GDCA with strategies did not perform well to solve instance A-03, B-05, and B-07 as six strategies cannot find a solution with less than 100% gap to the optimal solution. Table 4.2 shows a comparison of relat- ive gap between the best permutation order (see Permutation study) and the best ordering strategy. Only six instances have been used in this comparison because each of six instances has less than 5 sub-problems where all the per- mutation can be made. There are differences between the best strategies and the best permutation at maximum of 21.46%. Two out of six solutions (instance A-05 and B-02) of the best ordering strategy match the solution from the best permutation. This shows that the ordering strategies are able to work well in other problem instances.
The decomposition method is also able to find solutions for the large in- stances. The results from using the decomposition technique with the 6 order- ing strategies on the large instances are presented in Table 4.3. The table shows the objective values of the obtained solutions as relative gaps cannot be com- puted because the optimal solutions are not known. The values in bold are the lowest cost (best objective value) obtained among the six strategies. The table shows that as a whole, Desc-task gives six best solutions, Desc-ratio gives four best solutions, Asc-w&u gives two best solutions, Desc-w&u and Asc-task give
Table 4.3:Objective value obtained from solving large instances using six ordering strategies.
Instance Asc-task Desc-task Asc-w&u Desc-w&u Asc-ratio Desc-ratio D-01 1,688 496.45 1,549 765.48 1,301 240.98 D-02 860.50 372.94 496.47 495.44 984.98 732.97 D-03 2,625 3,213 2,619 3,837 1,691 3,839 D-04 312.43 418.89 303.45 283.91 314.42 420.41 D-05 408.42 243.89 1,113 253.91 401.45 241.89 D-06 307.55 1,411 946.60 1,583 634.05 1,729 D-07 1,113 753.28 292.55 604.01 293.53 1,077 F-01 73,287 64,305 71,430 72,040 75,761 63,681 F-02 81,853 73,291 76,460 80,570 86,906 74,860 F-03 141,060 115,235 140,258 120,715 148,092 116,011 F-04 111,671 102,994 105,262 109,411 113,557 91,670 F-05 127,476 101,438 113,403 105,284 112,995 103,156 F-06 105,595 76,007 88,702 84,050 107,281 84,050 F-07 199,160 176,541 194,525 178,387 218,059 178,387 Average 30,266 25,599 28,478 27,083 31,011 25,719
Boldtext refers to the best solution.
one best solution while the Asc-ratio gives no best solution. On average, the Desc-task strategy gives the lowest cost solution, around 17.45% less than the highest average cost strategy (Asc-ratio).
Finally, we use statistical test to validate our choice from the observation on the number of the best solution and the lowest average solution that Desc- task is the best ordering strategies for GDCA. Thus, Friedman ANOVA has been applied to measure the differences in objective values of between the six ordering strategies. Table 4.4 presents result from Friedman ANOVA which is in the form of in two sub-tables. The first sub-table shows the statistic value that
the calculated statistic value χ2 =11.335, the degree of freedom is 5, and the p-
value is .045. With significant level α =.05, the test shows that the mean ranks
between six ordering strategies are different significantly. The second sub-table presents the mean ranks of the six ordering strategies where the lower rank indicates the better solution. The mean rank confirms that Desc-task is the best ordering strategies amongst the proposed six methods as it has the lowest mean rank at 2.89. The highest mean rank ordering strategies is Asc-ratio where the value is 4.16. Therefore, in term of solution quality, we select Desc-task in the GDCA to compare with the other algorithms in Chapter 5 and Chapter 7 (full
Table 4.4:Friedman statistical test and mean ranks of objective value on six ordering strategies of GDCA. The lower mean rank presents better solution quality.
Friedman Test Mean Ranks
N 28 χ2 11.335 df 5 p .045 Feature Ordering Asc Desc task 4.11 2.89 w&u 3.05 3.41 ratio 4.16 3.38
comparison presented in Chapter 7).
Figure 4.4 shows, according to the problem size, the computation times used by the decomposition approach using the different ordering strategies and the time used to find the overall optimal solution. Each sub-figure presents the
problem instances classified by their size (number of items is |T| + |K|). Each
line represents the time used by the ordering strategy in solving the group of 14 problem instances. As noted before, the time to find the optimal solution
represented by is available only for the small instances. For the instances
which are smaller than instance B-06 (89 items), the computation time used by the decomposition method is not much different from the time used to find the optimal solution. The computation time used to find the optimal solution grows significantly for instances B-06 to B-03. The reason behind this is an increase in the problem size where the instances A-05 to B-04 have between 32 and 64 items while the four instances B-06, B-07, B-05, and B-03; have from 89 items to 103 items. Note that for instance B-03 which has 109 items, the MIP solver uses 5,419 seconds for finding the optimal solution. For the latter four instances, GDCA used less computational times than a half computational times of the MIP solver.
For the large instances, it is shown that the computation time used by the decomposition method starts from 17 minutes (1,060 seconds) to above 6 hours
A-05 A-07 B-02 A-04 A-06 A-02 A-01 A-03 B-01 B-04 B-06 B-07 B-05 B-03 0 100 200 300 400 500 5419
Instance (Order by #items)
Computation time (seconds) Small instances D-02 D-01 D-04 D-05 D-03 D-07 D-06 F-02 F-01 F-04 F-06 F-03 F-05 F-07 0.5 1 1.5 2 ·104
Problem size (items) Large instances
Asc-task Desc-task Asc-w&u Desc-w&u Asc-ratio Desc-ratio Optimal
Figure 4.4:Computation time (seconds) used in solving small and large in- stances. Each sub-figure corresponds to a problem size category (small and large). Instances are ordered by The problem size (#items) which is the summation of #workers and #visits. Each graph presents the computation time used by the decomposition method with the different ordering strategies (line with mark- ers) and the time used for producing the overall optimal solution (dashed line) when possible.
(22,478 seconds). Also, for the large instances the average computation time used by six strategies are 4,620 seconds; 3,098 seconds; 7,451 seconds; 6,348 seconds; 7,640 seconds; and 7,048 seconds respectively. The result shows the average processing time of Asc-task and Desc-task are significantly less compu- tation time than the other strategies. This is because these ordering strategies do not require an additional process to retrieve information about the problem. Again, we use Friedman statistical test to validate our computational time observation. Table 4.5 presents the result of the statistics in two sub-table. The first sub-table shows the statistic value of testing six strategies on 28 instances:
A, B, D, and F. The calculated value χ2 = 74.484, degree of freedom is 5, and
the p-value is less than .01. The Friedman test draws a conclusion that com- putational times between six ordering strategies are significantly different at
Table 4.5:Friedman statistical test and mean ranks of computational time on six ordering strategies of GDCA. The lower mean rank presents bet- ter solution quality.
Friedman Test Mean Ranks
N 28 χ2 74.484 df 5 p <.01 Feature Ordering Asc Desc task 2.21 1.46 w&u 4.91 4.07 ratio 4.48 3.86
dering strategies where the lower mean rank refers to the less computational time used. The result confirms that Desc-task is the fastest strategies with its mean rank at 1.46 and the second fastest strategies is Asc-task with the mean rank at 2.21. The other four strategies have very similar computational time where their mean ranks are between 3.86 to 4.91.
Hence, considering both solution quality and computation time, it can be concluded that Desc-task should be selected for large instances because it finds solutions which are overall the best in quality, provided by the objective value mean rank, and also which is the fastest ordering strategy, as shown in the computational time mean rank.