Evaluation and Computational Experiments
6.3 HACS based on spell-model
6.3.3 Evaluation of the swapping processes
In the tabu search algorithms of HACS, there are two main operations: swapping and recutting.
Once a set of spells are fixed, the swapping function attempts to improve the solution as much as possible by re-linking the units of vehicle work. The recutting function attempts to improve the solution by re-selecting AROs. Both functions play very important roles in refining the schedule. Their performances will affect the solution quality of HACS although the cooperation of all components of HACS determines the ultimate solution quality. This subsection intends to investigate the individual performance of the tabu search swapping algorithms; the recutting function is not independently tested because there is no appropriate way to judge its individual performance.
To evaluate the tabu search swapping algorithms, we design an experiment in which the initial schedule formed is based on a ‘good’ set of spells. The swapping fun ction, including all of the three swapping operations, is used solely to refine the schedule while the recutting-block and the adding-duties functions are switched off. The ‘good’ set of spells is taken from the best known solutions, i.e. TRACS II solutions.
• Forming initial solutions based on the spells in TRACS II solutions
The method of forming an initial schedule needs to be adjusted to use the spells in TRACS II solutions. In a TRACS II solution, some duties contain two-spells while the others may contain one, three or four spells (TRACS II only produces the duties with up to four-spells). However, HACS only initially forms the duties with up to two-spells. The number of duties in an initial schedule would be larger than that in the TRACS II solution containing three or four spell duties. Although it would be possible for the swapping process to reduce the number of duties
used, it is anticipated that the current HACS would have difficulty in reducing many duties. To focus on the performance of the tabu search swapping algorithms, the non-two-spell duties are adopted into the initial schedule while the spells in the other duties are re-coupled to form new two-spell duties. We use two different methods to couple the spells in order to test the swapping algorithms from different starting solutions. One method is the HACS’ initialisation method.
Another method forms an initial schedule by first sorting the spells by starting time, and then coupling two adjacent spells. The duties generated by one of the methods plus the original non-two-spell duties constitute an initial schedule. The HACS approach modified to test the swapping algorithms is denoted as HACS-1 or HACS-2 respectively according to the initialisation method incorporated.
• Experiment 1
The problem D6 was first selected for this experiment because the RPD, in terms of number of duties, over the best known solution is the highest amongst the test data listed in Table 6.3. It was expected that a better solution would be produced by using the ‘good’ set of spells in the best known solution. The best known solution to D6 costs 363 hours and 26 minutes and contains 39 duties, but only 18 of which are two-spell duties while the other duties are three-spell duties. Based on the 36 three-spells included in the 18 two-three-spell duties, two starting solutions and the corresponding final solutions are generated, which are summarised in Table 6.5, in which the similarity to the best known solution is analysed. In the initial solutions, the number of non-two-spell duties is presented in the first column, which is constant in both methods.
When a solution is infeasible, the penalty of which is presented with ‘p’ following the number.
The cost of a feasible solution is followed by a ‘c’.
Table 6.5: The starting solutions and the corresponding final solutions to D6 obtained by HACS based on the spells in the TRACS II solutions.
Initial Schedule Final schedule
From Table 6.5, we can see that HACS-1 produces a solution 3 minutes cheaper than the best known solution starting from a more expensive feasible solution, and HACS-2 does not produce a feasible solution starting from a very poor initial solution. However, HACS-2 has improved the initial schedule considerably. In the initial schedule of HACS-2, every newly generated duty is illegal and different from the original duties. After swapping, most of the duties are still different. In HACS-1, the number of different duties has increased after swapping although the total cost is very close. The results indicate that there may be many possible combinations to form good feasible solutions based on the same set of spells.
• Experiment 2
In the previous experiments, the best known solution only contains 46% two-spell duties. This has reduced the difficulty to form a good initial schedule for HACS-1 because 54% of the duties are already formed and are valid. That case is not very common; in many cases two-spell duties are the majority in a solution. In this experiment, we select D7 as the test problem. The best solution to D7 has 408 hours and 47 minutes in cost and contains 49 duties, 48 of which are two-spell duties while only one is a three-spell duty. HACS-1 forms an initial schedule containing two infeasible duties. This initial schedule differs greatly from the best known solution. After swapping, a feasible solution is obtained, which is 48 minutes more expensive
and 55% of the duties are different from those in the best known solution. HACS-2 forms an extremely poor initial schedule, from which a considerably better solution is obtained after swapping although it is still infeasible. The details are shown in Table 6.6.
Table 6.6: The starting solutions and the corresponding final solutions to D7 obtained by HACS based on the spells in the TRACS II solutions.
Initial Schedule Final schedule
Similar to the previous experiment, this experiment shows that the HACS initialisation method produces a better initial solution than the method in HACS-2, and the better initial solution leads to a better final solution.
• Experiment 3
This experiment selects D9 as the test problem, which is the second largest amongst the test problems. The best known solution to D9 has 632 hours and 32 minutes in cost and contains 80 duties, 54 of which are two-spell duties. HACS-1 forms an initial schedule containing 8 infeasible duties. This initial schedule differs greatly from the best known solution. After swapping, a feasible solution is obtained, which is 1 hour and 56 minutes more expensive.
HACS-2 forms an extremely poor initial schedule, from which a considerably better solution is obtained after swapping, which contains two invalid duties with 40 minutes penalty. The details are shown in Table 6.7. In this experiment, the solutions obtained by HACS-1 and HACS-2 are considerably different from the TRACS II solution, and the number of different duties is not explicitly counted.
Table 6.7: The starting solutions and the corresponding final solutions to D9 obtained by HACS based on the spells in the TRACS II solutions.
Initial Schedule Final schedule
The foregoing experiments show that the HACS-1 solutions to the three problems are of a quality comparable to the best known solutions. Hence, the tabu search swapping algorithms have reached an acceptable level of performance because we do not expect an optimal solution.
As for the HACS-2 solutions, which are not good enough, the tabu search swapping algorithms do improve the extremely poor starting solutions considerably. For local search methods, better initial solutions usually lead to better final solutions.
6.3.4 Summary
This section has presented a series of experiments on the HACS approach with the spell-model.
The experiments have shown the importance of multi-neighbourhood structures and the tabu search framework. The multi-neighbourhood structures used in combination are superior to a single neighbourhood structure used in isolation, and the tabu search technique is superior to the steepest descent method. Compared with TRACS II, HACS can produce solutions considerably quicker and is very easy to manipulate because many parameters in TRACS II requiring careful setting are not needed. The experiments on solely evaluating the swapping function have shown that the swapping function has reached an acceptable level of performance. Given a good set of spells, HACS could generate a good solution. However,
without this condition, the HACS solutions may be unsatisfactory. The next section will focus on the enhancement of the recutting function by incorporating the piece-model.