TRACS II solutions to the TEST problem and analyses

It has already been mentioned that none of the existing systems can directly handle windows of relief opportunities. There are only two possible ways for TRACS II to tackle the TEST problem. They are discussed separately below.

1) Shrink a WRO into a RO by selecting only the arrival time as a relief time

This is a common method for the existing approaches to solve practical driver scheduling problems. After the change, the blocks included in the TEST problem were converted into those as displayed in Figure 6.2, where no WROs existed.

0812 0950 1152 1332 1455 1655 vehicle 1

G H H H H G

1240 1330 1430 1550 1710 1855 vehicle 2

G H H H H G

0920 1112 1230 1350 1505 vehicle 3

G H H H G

TRACS II was then run and produced a solution as shown in Figure 6.3. This solution contained four duties with a total wage cost of 24 hours and 58 minutes, which was worse than the HACS solution in the terms of both the number of duties and the cost.

Figure 6.2: Blocks in the TEST problem after shrinking each WRO into a RO

0812 0950 1152 1332 1455 1655 vehicle 1

G H H H H G +---Duty 1---+---Duty 4---+---Duty 2---+

1240 1330 1430 1550 1710 1855 vehicle 2

G H H H H G +---Duty 1---+---Duty 3---+

0920 1112 1230 1350 1505 vehicle 3

G H H H G +---Duty 2---+---Duty 3---+

Comparing the TRACS II solution with the HACS solution, we found that the TRACS II solution contained seven spells while the HACS solution had six. Further investigation found that TRACS II partitioned the first block into three spells while HACS divided it into two spells. The explanation is: the length of the first block is 8:43, which is longer than the maximum length of spell (five hours). The block therefore has to be partitioned into at least two spells. The longest early spell could be from 8:12 to 13:12 (the sum of 8:12 plus 5:00).

However, since 13:12 is not a relief time, an earlier relief time 11:52 was selected. Similarly, the longest spell covering the end of the block could be from 11:55 to 16:55. Since 11:55 is not a relief time, the later time 13:32 was selected. The work between 11:52 and 13:32 was not covered; hence, three spells were essential to cover all the work in the first block.

Figure 6.3: TRACS II solution to the TEST problem after shrinking each WRO into a RO

However, in the original TEST problem presented in Figure 6.1, the time 11:55 in the first block was a relief time falling in the time window from 11:52 to 11:57. It was possible to partition the block into two at the time 11:55, 11:56, or 11:57. The WROs enabled the use of one fewer spell and to produce the better solution. This experiment has exemplified the improvement that WROs may bring and that shrinking WROs might lose some opportunities to produce a better solution.

2) Expand explicitly each WRO into a sequence of individual ROs at one-minute intervals

This method would not normally be practical because it will increase considerably the number of ROs, and is thus not suitable for the generate-and-select based approaches, e.g. TRACS II, in which the magnitude of a problem is largely dependent on the number of ROs. Since the TEST problem is artificially tiny in terms of vehicle work to be covered, it would be worth trying.

Two experiments were carried out: one used slack filtering rule parameters while another used tight parameters.

Theoretically, using slack filtering parameters enlarges the search space providing more chance of producing a good solution. HACS does not need any filtering rules because it only refines the small solution set of duties constructed. However, TRACS II has to apply filtering rules so that the set of potential duties generated would not be prohibitively large. In the duty generation process, TRACS II first generates the set of potential legal spells, the length of each is limited in a time range stipulated by parameters. It then generates the set of potential legal stretches based on the set of spells. Finally by combining the stretches a large set of potential duties is generated. There are upper bound limits for the number of potential spells, stretches generated.

To run TRACS II in a similar condition to HACS, slack filtering rule parameters were used in the first experiment. Unfortunately, TRACS II terminated before finishing the duty generation process because the number of stretches exceeded TRACS II’s limit (899 spells and 26637 stretches had been formed before the termination).

The second experiment was then designed using tighter parameters in order to reduce the number of stretches to be formed. The minimum lengths of spells and stretches were set to one hour and three hours respectively. A solution was obtained in about ten minutes, which contained four duties and the total cost in hours and minutes was 23:53. Figure 6.4 demonstrates the schedule, where the times underlined are active relief times. Duty 1 is a straight duty with a meal break and the other duties are non-mealbreak duties, amongst which duty 4 contains two spells.

Figure 6.4: TRACS II solution to the TEST problem after expanding

In the foregoing experiment, WROs were represented in the same way as in HACS, but the solution contained one more duty. The reason was investigated below.

Comparing the solutions, we found in the HACS solution that the shortest stretch, which contained only one spell, was 1 hour and 55 minutes. This stretch was shorter than the

minimum length of stretch stipulated in TRACS II. Therefore, it was impossible for TRACS II to produce such a solution.

6.5.3 Summary

This section has demonstrated that HACS can take advantage of WROs, but the generate-and-select approaches, e.g. TRACS II, cannot because shrinking WROs or using tight parameters compromises the quality of solutions while the slack parameters cause the set of potential stretches or duties generated to be too large even for trivial problem instances with WROs. It is impractical to expand explicitly each time window into individual minutes using the existing generate-and-select approaches.