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Overtime shifts consist of one-part shift of around two to four hours. These are in addition to normal shifts for relatively high payment and may be useful in covering
peaks in demand.
2.3.2 Heuristics
Before the 1980s, heuristics were mainly used to solve crew scheduling problem because computers ware not powerful enough to run mathematical programming
approaches, and the mathematical programming techniques were also not advanced Wren and Rousseau, 1995. A complete review of heuristics can be found in Wren and
Rousseau 1995 and Wren 1998. Heuristic approaches rely on the knowledge of expert schedulers to build schedules, and many approaches start by constructing an
initial schedule and then improving the schedule by making limited alterations. Examples of heuristic based approaches are TRACS Techniques for Running
Automatic Crew Scheduling, which was developed at the University of Leeds Parker and Smith, 1981; RUCUS Run Cutting and Scheduling in America by The Mitre
Corporation Bennington and Rebibio, 1975;Bodin et al., 1985; HOT Hamburger Optimierungs Technik in Germany Hoffstadt 1981; Daduna and Mojsilovic, 1988
and COMPACS COMPuter Assisted Crew Scheduling at the University of Leeds Wren et al., 1985
According to Wren 1998, the approach taken by TRACS is first to create an initial schedule that satisfies all the labour agreement constraints, then a set of refining
heuristics is applied to the schedule generated in order to try to improve its quality both in terms of number of duties and costs. An initial schedule is constructed in the
following steps: form early duties from the beginning of the bus schedule; construct late and middle duties at the end of the bus schedule; put together split duties by matching
early duties with late or middle duties; attach remaining work to existing duties if possible.
This initial schedule contains two-spell and three-spell duties. A concept of marked time is used to guide the formation of duties. For early duties, a marked time is the
latest time by which the first crew of each bus must be relieved. For late duties, a
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marked time is the earliest time at which the last crew can start work on each bus. To minimise the number of crews during peaks hours, meal break chains are designed so
crews can take meal breaks in turn Fores et al., 1998; Wren et al., 2003. The initial schedule is improved by two sets of procedures Fores et al., 1998. The first
set attempts to reduce the number of duties by considering every duty and whether it can be fitted in with other duties. This procedure redistributes work between duties so
that duties with long spreadovers are assigned with more work. This makes short duties become shorter and they subsequently can be removed. The second set is intended to
reduce costs by implementing several procedures including swapping or moving parts of duties, re-matching first and second parts of stretches of duties, switching the relief
point of a duty to another relief time, and relocating short pieces of work around the middle of the day.
Parker and Smith 1981 reported that TRACS was implemented in the 1970s for several UK bus companies with some success. These include Bristol Omnibus
Company, Midland Red, Cleveland Transit, West Yorkshire Passenger Transport Executive, Greater Glasgow Passenger Transport Executive and Great Manchester
Passenger Transport Executive. However, Parker and Smith 1981 observed that modifying the programme to work with different labour agreement rules could be
difficult. RUCUS was developed in the late 1960s. According to Bodin et al. 1985 RUCUS
first generates an initial solution and then refines it using heuristics. First, one spell duties are formed then two spell duties, finally, any unallocated pieces of work are
considered as overtime duties. After the initial schedules are ready, the system begins to use local search to improve the schedules. It either exchanges parts of work with other
duties, or moves selected relief points forward or backward. A repair procedure is then used to fix any unfeasible duties due to the changes. In case there are still unfeasible
duties left in the final schedules, manual interventions are used. HOT was developed and has been used by schedulers at Hamburger Hochbahn AG
since the 1970s Daduna and Mojsilovic, 1988. It starts by trying to form good duties,
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one at a time, for early buses, and then for late buses. Any pieces of works that cannot fit into the initial schedules are formed into partial duties, which are then combined into
full duties by a variant of the Hungarian Algorithm Taha, 1997. Little improvement can be achieved to the schedule once it is constructed, and sometimes it may even leave
unassigned pieces of work Wren and Rousseau, 1995. COMPACS COMPuter Assisted Crew Scheduling is an interactive system developed
in the early 1980s Wren et al., 1985, and later incorporated into the BUSMAN scheduling package Chamberlain and Wren, 1992. According to Wren et al. 1985
COMPACS is designed to operate in two different ways. One way is to produce an entire duty schedule automatically using heuristics similar to those used in TRACS. The
schedule is then created quickly with no interaction from the user. The second produces a duty schedule in an interactive fashion. Users can create their own duties to add to the
schedule or ask COMPACS to form duties, which can then be accepted or rejected as seen by the scheduler. At any point the scheduler can have COMPACS finish the
partially created schedule automatically. The scheduler also has complete control over the schedule through facilities to edit or delete duties, which are already present in the
schedule.
2.3.3 Mathematical Programming
