For these 8 days, each algorithm was run 10 times per initial construction method and 75 minutes per run. Table 6.5 shows the results of these test runs. Depending on the construction method (CP or Random Construction), each improvement algorithm uses the same initial solution. The value in the 3. column shows the objective value of the initial solution. For each algorithm (starting from column 4) the average best solution is provided as well as the standard derivation (below). The VND results have been obtained by using the objective value after the first VND call in the VNS. Note: instance 3 (day 3) has an error in the data which makes it impossible to create a valid solution.
As table 6.5 shows, the MA approach obtained the best results over all tested instances, whether using the CP or random initialized solution. Also in terms of robustness the MA per-forms best compared to the other approaches as it produces very similar results in terms of the objective value.
38
Table 6.5:Final objective values for each metaheuristics, averaged over 10 runs
constr. init. VND VNS EA SAHH SS
day 1 cp 0.0885 0.03080 0.03038 0.02744 0.03260 0.03097
±0.00067 ±0.00034 ±0.00057 ±0.00090 ±0.00034
rand. 162.1513 0.33083 0.33064 0.02707 0.03257 4.03197
±0.48339 ±0.48340 ±0.00031 ±0.00058 ±0.00065
day 2 cp 0.0894 0.03090 0.03038 0.02794 0.03381 0.03120
±0.00039 ±0.00028 ±0.00037 ±0.00081 ±0.00050
rand. 152.1521 0.53162 0.43109 0.02767 0.03321 3.03098
±0.52725 ±0.51647 ±0.00037 ±0.00082 ±0.00055
day 3 cp 1.0860 1.02879 1.02837 1.02600 1.03063 1.02931
±0.00055 ±0.00044 ±0.00051 ±0.00096 ±0.00051
rand. 148.1494 1.02895 1.02851 1.02553 1.03023 2.02875
±0.00043 ±0.00040 ±0.00036 ±0.00070 ±0.00048
day 4 cp 0.0896 0.03132 0.03083 0.02874 0.03302 0.03220
±0.00044 ±0.00048 ±0.00042 ±0.00073 ±0.00051
rand. 155.1532 0.03232 0.03188 0.02784 0.03470 0.03069
±0.00083 ±0.00087 ±0.00056 ±0.00113 ±0.00040
day 5 cp 0.0885 0.03117 0.03076 0.02850 0.03283 0.03202
±0.00052 ±0.00050 ±0.00044 ±0.00083 ±0.00077
rand. 168.1506 0.03167 0.03131 0.02805 0.03286 0.03106
±0.00057 ±0.00068 ±0.00047 ±0.00061 ±0.00048
day 6 cp 0.0867 0.03028 0.02964 0.02665 0.03200 0.03057
±0.00055 ±0.00060 ±0.00048 ±0.00080 ±0.00068
rand. 145.1523 0.03040 0.02992 0.02646 0.03228 0.02947
±0.00051 ±0.00056 ±0.00025 ±0.00076 ±0.00043
day 7 cp 0.0904 0.03209 0.03165 0.02947 0.03377 0.03300
±0.00049 ±0.00054 ±0.00035 ±0.00098 ±0.00050
rand. 149.1516 0.03270 0.03218 0.02900 0.03318 1.03271
±0.00085 ±0.00080 ±0.00042 ±0.00062 ±0.00049
day 8 cp 0.0872 0.02914 0.02846 0.02576 0.02994 0.02995
±0.00071 ±0.00056 ±0.00038 ±0.00061 ±0.00046
rand. 158.1446 0.22922 0.22893 0.02520 0.02969 4.46026
±0.63228 ±0.63227 ±0.00041 ±0.00064 ±0.53455
The VNS approach implemented by the AIT got the second best scores but only if started with a valid solution, i.e., a solution constructed using CP. For day 1, 2 and 8 the VNS failed to find valid solutions using a randomly initialized solution for at least one run.
These instances seem to be difficult to solve using only a randomly generated solution as for them and the day 7 instance the scatter search as proposed in this thesis could not find a valid solution on any run. The results of the scatter search are similar to those of the VNS except for the mentioned instances using random starting solutions.
From the results of the SAHH it can be observed that this approach may not find solutions as good as the other approaches, but it is independent of the starting solution and could find a solution for the previously mentioned difficult instances every run.
More information on the behaviour of the metaheuristics can be obtained by comparing the changes of the objective value with the changes of the average travel time and the number of nurses during the search. These values have been taken from intermediate solutions of the search procedures using the best run of the day 1 instance with either the CP (6.1a,6.2a,6.2c) or the random generated solution (6.1b,6.2b,6.2d). Note: The VNS values shown in the figures are obtained every iteration of the VNS, i.e., not every step of the embedded VND is plotted.
Objective value development over time on Day1 (CP)
Time (secs)
Objective Value
1 1000 2000 3000 4000 5000
0.020.050.2110100
Objective value development over time on Day1 (Random)
Time (secs)
Objective Value
1 1000 2000 3000 4000 5000
0.020.050.2110100
Figure 6.1:Changes in the objective value over runtime for day1 for CP (a) and random (b) initialization.
The changes of the objective value (6.1a,6.1b) show that the MA finds a solutions better than obtained by the other approaches early in the search and is able to improve it further. As it can be seen in figure 6.2a the number of nurses increases slightly while the average tour length (Figure 6.2c) decreases at the same time. Therefore it can be argued that first improvements of the initial solution generated by the CP can be obtained by increasing the number of nurses and decreasing the travel time. More interesting are the further improvements made by the MA. By decreasing the number of nurses below the number of the initial value while still having travel times also below the initial one the MA can outperform the other approaches.
A similar observation can be made when looking at figures 6.2b and 6.2d for the randomly initialized solution. Here the first improvements are made by optimizing the travel times and reducing the number of nurses at the same time (as, by construction, all nurses have at least one job assigned). After the initial decrease of the average travel time it increases again while the number of nurses still decreases.
Both observations suggest, that a good solution based on the objective function used in this thesis is more characterised by a low number of nurses used than on a low average travel time per nurse.
The changes in the objective value of the approaches for the other instances in this compari-son are shown in figures 6.3, 6.4, 6.5 and 6.6. These show a similar behaviour as the discussed results of the day 1 instance. The MA and the SAHH converges early to the region of very good solutions but only the MA is able to further decrease the objective value thus leading to superior results.
With respect to the number of nurses used within the solution, it can be observed that all approaches reach solutions where the final number of nurses employed is relatively low (see table 6.6). Such solutions are preferred as future work will consider solving multi day instances 40
0 1000 2000 3000 4000 5000
150200250300350400450500
#Nurses in the solution over time on Day16 (CP)
Time (secs)
0 1000 2000 3000 4000 5000
150200250300350400450500
#Nurses in the solution over time on Day16 (Random)
Time (secs)
0 1000 2000 3000 4000 5000
50556065
Travel Time in the solution over time on Day16 (CP)
Time (secs)
Average Travel Time Per Tour (5min)
Metaheuristics
0 1000 2000 3000 4000 5000
50556065
Travel Time in the solution over time on Day16 (Random)
Time (secs)
Average Travel Time Per Tour (5min)
Metaheuristics
Figure 6.2: Key indicator performance over runtime for day1: number of used nurses for CP (a) and random initialization (b); average travel time per nurse for CP (c) and random initialization (d).
which have additional constraints for employed working days. Thus a low number of nurses will provide an additional grade of flexibility when refining a day’s schedule.
Objective value development over time on Day2 (CP)
Time (secs)
Objective Value
1 1000 2000 3000 4000 5000
0.020.050.2110100
Objective value development over time on Day2 (Random)
Time (secs)
Objective Value
1 1000 2000 3000 4000 5000
0.020.050.2110100
Objective value development over time on Day3 (CP)
Time (secs)
Objective Value
1 1000 2000 3000 4000 5000
0.020.050.2110100
Objective value development over time on Day3 (Random)
Time (secs)
Objective Value
1 1000 2000 3000 4000 5000
0.020.050.2110100
Figure 6.3:Changes in the objective value for day 2 with CP (a) and random initialization (b); and day 3 with CP (c) and random initialization (d). Note that day 3 has an error in the instance data making it impossible to create a valid solution.
42
Objective value development over time on Day4 (CP)
Time (secs)
Objective Value
1 1000 2000 3000 4000 5000
0.020.050.2110100
Objective value development over time on Day14(Random)
Time (secs)
Objective Value
1 1000 2000 3000 4000 5000
0.020.050.2110100
Objective value development over time on Day5 (CP)
Time (secs)
Objective Value
1 1000 2000 3000 4000 5000
0.020.050.2110100
Objective value development over time on Day5 (Random)
Time (secs)
Objective Value
1 1000 2000 3000 4000 5000
0.020.050.2110100
Figure 6.4:Changes in the objective value for day 4 with CP (a) and random initialization (b); and day 5 with CP (c) and random initialization (d).
Objective value development over time on Day6 (CP)
Time (secs)
Objective Value
1 1000 2000 3000 4000 5000
0.020.050.2110100
Objective value development over time on Day6 (Random)
Time (secs)
Objective Value
1 1000 2000 3000 4000 5000
0.020.050.2110100
Objective value development over time on Day7 (CP)
Time (secs)
Objective Value
1 1000 2000 3000 4000 5000
0.020.050.2110100
Objective value development over time on Day7 (Random)
Time (secs)
Objective Value
1 1000 2000 3000 4000 5000
0.020.050.2110100
Figure 6.5:Changes in the objective value for day 6 with CP (a) and random initialization (b); and day 7 with CP (c) and random initialization (d).
44
Objective value development over time on Day6 (CP)
Time (secs)
Objective Value
1 1000 2000 3000 4000 5000
0.020.050.2110100
Metaheuristics EA VNS SS SAHH
(a)
Objective value development over time on Day6 (Random)
Time (secs)
Objective Value
1 1000 2000 3000 4000 5000
0.020.050.2110100
Metaheuristics EA VNS SS SAHH
(b)
Figure 6.6:Changes in the objective value for day 8 with CP (a) and random initialization (b).
Table 6.6:Number of nurses finally scheduled for each day.
day 1 day 2 day 3 day 4 day 5 day 6 day 7 day 8
VNS 200 205 212 208 204 202 216 211
MA 175 178 175 178 178 174 183 180
SAHH 202 206 213 202 203 203 216 204
SS 201 206 213 212 210 212 216 208
CHAPTER 7
Conclusion
In this thesis three metaheuristics were implemented tackling a multimodal home healthcare scheduling problem based on the working procedures of a Viennese home healthcare company.
A detailed description of the problem and solution as well as the solving architecture for the single-day instance was provided. The algorithms were selected for their performance on related problems, i.e., the vehicle routing problem with time windows and the nurse rostering problem, and adapted to the MHS.
First a simulated annealing hyper heuristic was adapted by using the already existing neigh-bourhoods as low-level heuristics and experimenting with the parameters and temperature val-ues. Also a constrained variant for the random improvement strategy was proposed but discarded after preliminary results indicated no improvement compared to the unconstrained variant. The results show the good performance in comparison with the existing VND and VNS approaches.
As a second approach a memetic algorithm was implemented which was redesigned early to a steady-state approach using a tailored recombination operator. Another improvement was obtained using a cyclic search of the neighbourhoods as embedded local search procedure. As the results of the final comparison showed, this approach performs best compared to the other proposed metaheursitics.
The last approach implemented was a scatter search. After first tests, theconstruction by votingapproach was observed to not be suitable to tackle the problem, thus a path-relinking like algorithm was proposed which improved the performance considerably. However, even with this improvement and after experiments with different parameters and local search procedures, this search performed worst of all compared approaches when using a randomly initialized solution, as the search was not able to find valid solutions on each run of 4 out of the tested 8 instances.
Future work could consist of using these metaheuristics to solve multi-day solutions. This could be realised using the proposed algorithms in a repeating 2-phase approach where solutions are created per day and refined by adapting the objective function. Of all presented approaches, the memetic algorithm might perform very well considering its ability to find good results early in the search with a low number of nurses used. Furthermore the population maintained by
the MA could be used to refine a multi-day solution without the need to start a new search procedure.
48
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