Análisis descriptivo

TRACS II (Techniques for Running Automatic Crew Schedules, Mark II) has been developed since 1994 specifically to satisfy the needs of rail operations, while still being capable of producing bus driver schedules. TRACS II follows almost the same approach as IMPACS, but the components have largely been redesigned to cope with the complexity of rail operations and to incorporate new algorithmic advances.

From Figure 2.1, it can be seen that TRACS II consists of seven modules in three stages. The first stage is called shift generation, which constructs many potential duties. VALIDATE

Figure 2.1 Workbench main window of TRACS II

checks first the validity of data sets. Then TRAVEL calculates all possible opportunities for drivers to travel as passengers on scheduled services or taxis between relief opportunities.

BUILD finally generates a large set of possible valid duties (a set of filtering parameters is used to limit the generation of too many duties). The heuristics in BUILD are different from IMPACS. The second stage contains a module called SIEVE, which ranks the duties generated by BUILD and eliminates those deemed to be inferior. The set of duties generated by BUILD is trimmed down to a desired size before it is passed to the mathematical solver called SCHEDULE. Recently, SIEVE is often only used to remove duplicated duties because the integration of a column generation method into SCHEDULE now allows it to accept larger duty sets. MERGE is only used when BUILD is run more than once. It merges sets of duties generated according to different parameters or different decomposed sub-problems. The mathematical process called SCHEDULE is the main module in the third stage, which is originated from the ILP process used in IMPACS. IMPACS could handle up to 30,000 input duties, but there is virtually no limit for TRACS II. SCHEDULE has two main choices of optimisation routines, which are that of a primal column generation approach and a dual steepest edge approach. Fores et al (2001) recommended that the dual steepest edge approach be executed on problems containing at most 30,000 duties, where all duties can be explicitly considered within the limits imposed for storage and efficiency. However, users can select the column generation approach to run any size of problem. SCHEDULE consists of four processes. The first process builds an initial integer solution using heuristics if column generation is selected as the solution algorithm. This process also selects an initial duty set of up to 30,000 duties from the large set of potential duties generated by BUILD. The second process uses a set covering model and obtains a LP relaxation. The second process will be repeated if it is necessary to add new columns of duties. The third process called REDUCE is introduced by Smith and Wren (1988) and enhanced by Willers (1995). It selects the duties that start or end at an RO that is used in the LP solution. The other duties are removed and pieces of

work not using a selected RO are combined. This is an optional process, but reduction in problem size decreases solution time and does not seem to inhibit the quality of the schedule.

The fourth process searches for an integer solution using a Branch and Bound method. If a solution is found, DISPLAY will print the schedule in a condensed format.

TRACS II has been successfully installed in several transport companies. Over the period 1995-1996 TRACS II satisfactorily produced schedules for about twelve of the privatised British train companies (Kwan et al., 1999). It is now being installed in the 26 companies of First Group, the largest bus company in the UK.

TRACS II is also used for benchmarking by a number of researches in Leeds University. Parts of the system are utilised in some other driver scheduling approaches, e.g. Forsyth’s ant system, Kwan’s and Li’s genetic algorithms, Li’s simulated evolution algorithm, and La yfield’s and Curtis’s constraint programming approaches (see Sections 2.4 and 2.5).

Since TRACS II uses a set covering model, the schedule generated may (and usually) contains overlapping driving work (over-cover), which creates unproductive time for drivers. Over cover is eliminated manually before a schedule is operated.

Also, because of the use of ILP, a series of techniques to reduce the number of variables (potential duties) and constraints (ROs) have to be operated according to the capability of the ILP solver. Figures 2.2 and 2.3 illustrate the processes of reduction, where A, B, C, D denote the sets of relief opportunities at different levels, while E, F, G, H, I denote the sets of duties at different levels. The reduction at each level decreases the possibility of producing an optimal solution.

A: Full set of windows of relief opportunities.

B: Reduced set of relief opportunities after shrinking a WRO into one or two selected ROs.

C: Reduced set of relief opportunities after removing the ROs not used in LP relaxation.

(This is optional so might not be reduced).

D: Set of active relief opportunities.

E: Set of all possible valid duties according to labour agreement rules

F: Set of all possible valid duties based on tightened governing rules and the reduced set of ROs in level B, i.e. not considering windows of relief opportunities.

G: Reduced set of potential duties selected by the column generation method.

H: Reduced set of potential duties selected based on the set of ROs in level C.

I: Set of duties in a schedule, which may contain overlapping duties.