Railway crew scheduling is governed by a great variety of legal rules and regulations of public transport authorities (e.g., the Federal Railway Administration (FRA) in the US, see Kumar et al. (2009)), complex labor union contracts (see Abbink et al. (2005), J¨utte/M¨uller/Thonemann (2017)), individual operating conditions and further rules defined by the operator. Researchers in RCSP, therefore, are obliged to build an extensive knowledge about the instance-specific rules and restrictions and to accurately translate them into a mathematical model. To manage the size of the solution space, it is common practice to relax constraints or differentiate between hard and soft rules, which are frequently modeled as constraints and penalties, respectively (see Abbink et al.
3.4 Model formulations, objectives and constraints of RCSP 49
(2005), Banihashemi/Haghani (2001), Khmeleva et al. (2018)).
For the sake of brevity, we refrain from surveying and categorizing all instance-specific forms of constraints in detail. For country and operator specific rules and restrictions, we refer the reader to publications of country cases, which can be seen in Table 3.2. Here, we give an overview of the most common constraints and highlight selected examples of special constraints or model extensions. In general, constraints apply to different levels of a schedule: to the duty itself, to a depot, to the total schedule or to specific types of crew members.
Duty-based constraints must be fulfilled to create a legal duty. This includes funda- mental constraints such as a geographical and chronological link between two consecutive trips and the same start and end depot per duty (known as duty symmetry). In some cases, it appears useful to restrict the number of trips per duty to a maximum (see Froger/Guyon/Pinson (2015), Park/Ryu (2006)) or to define a maximal trip du- ration (see Shen/Chen (2014)). The minimal connection time between train changes needs to reflect times for walking, handover tasks, etc. and has a major effect on the robustness of the schedule (see Section 3.4.2). Deadheading by train is often allowed in RCSP, even other transportation modes such as taxi can be used (e.g., see Janacek/ Marton/Koniorczyk (2016)). Abbink et al. (2005) add penalties to the objective function in order to reduce the number of such positioning trips. In freight operations and special cases of long-distance passenger train services, the away rest time during lay- overs and the home rest time between duties have a lower and upper bound (see Muroi/ Nishi/Inuiguchi (2010), Kumar et al. (2009)). Most restrictions on duty-level refer to time aspects in order to ensure sustainable working hours for crews. This includes a maximal consecutive or total driving time (or distance) and a minimal and maximal total working time (or duty duration). Besides driving time, the latter includes break times, idle times and all non-driving tasks, in particular additional sign-on/-off time or preparation/finishing tasks such as quality checks, paperwork and handover meet- ings (see Freling/Lentink/Odijk (2001), S¸ahin/Y¨uceoˇglu (2011)). If the total working time does not equal the total paid time, a minimal paid time per duty can be required (see Hoffmann et al. (2017)). Break rules belong to the most complex regu- lations. They usually depend on the duty duration. For instance, in the Netherlands, duties longer than 5:30 hours require a meal break of at least 30 minutes (see Kroon/ Fischetti (2000)). Hence, the number and duration of breaks must fulfill a certain minimum and they can also be restricted to a maximum at the same time. It is also common to define the latest possible start time of a break, usually after a certain time of duty duration (e.g., meal breaks must start at the latest after 5:30 hours). Han/Li (2014) and Zhou et al. (2016) allow breaks only at pre-defined time windows during the
day. It should be noted that breaks can only occur at defined relief points with meal break opportunity (see Muroi/Nishi/Inuiguchi (2010)).
At depot-level and schedule-level, a subset or the total set of duties in the solution are subject to fulfill certain constraints. Commonly, railway networks exist of multi- ple depots from which crews might operate. Problems with a single depot are rather rare, they occur, for instance, when only one line is scheduled. The number of duties or crews assigned to each depot might be restricted (known as depot capacity). The average working time, duty duration or paid time can apply at depot-level (see Kroon/ Fischetti (2000)) or to the total schedule (see Shen/Chen (2014), Hoffmann et al. (2017)). Similarly, specific duty types such as night duties or part-time duties with spe- cial break time rules might apply. Their balancing across depots or weekdays increases the problem’s complexity (see e.g., Shen/Chen (2014)).
It is common practice to schedule each type of crew (driver, conductor, catering staff, security guard) separately because of variation in rules and restrictions (Kwan (2011)). Line or traction knowledge is very important for drivers; they usually need a qualification for operating a specific vehicle type (see Abbink et al. (2005), Dornberger/Frey/ Schmid (2007), Khmeleva et al. (2014), Kumar et al. (2009), Kwan/Kwan/Wren (1999)). For conductors, Hoffmann (2017) considers minimal attendance rates below 100% per line, day-time or other categories. Snijders/Saldanha (2017) address, among others, the one-or-nothing behavior of duties for security guards on trains, i.e. one trip of the train is covered or the train is not covered at all. Kumar et al. (2009) and Vaidyanathan/Jha/Ahuja (2007) include the assignment of crews in FIFO (first in - first out) order at a depot. For safety reasons, Shen/Chen (2014) restrict the time period on the same type of train and S¸ahin/Y¨uceoˇglu (2011) add a required double manning time.