Previous Work

In the area of transport scheduling and routing, the concept of a ‘time window’ is usually associated with a constraint that some action (e.g. a pick-up or delivery, arrival or departure) must happen within certain time bounds; in this sense, adding the consideration of time windows is usually a way of specializing a problem by adding new constraint dimension to it. Although WROs as introduced in the previ- ous chapter could be seen as delimiting a range of times in which a certain action can happen, we observe that driver scheduling with WROs is instead a generaliza- tion of the original driver scheduling problem, as described in Section 1.3. From a methodological point of view, we are interested in comparing TDS and TDSW, and investigate the benefits in considering WROs. It is then important to differentiate research in time windows in terms of whether they restrict or expand the original problem.

Time windows have been studied for a long time in the area of vehicle routing, although mostly in a restrictive sense. Time windows in daily aircraft routing and scheduling are considered by Desaulniers et al. [21]. Two models are proposed: a set- partitioning one, which is solved using branch-and-bound and column generation, and a time-constrained multicommodity network flow formulation. Their column generation approach also involves the use of a multicommodity network model for the subproblem. Multicommodity network flow formulations have remained popular in vehicle scheduling [6, 85]; however, they do not translate easily to the crew scheduling problem.

Incorporation of time windows as an extension to vehicle or crew scheduling is more recent. Klabjan et al. [51] propose a variation to the crew scheduling problem in airlines, where the flight departure times are allowed to be modified within given time windows for the purposes of obtaining a better crew schedule. Although the problem is different than the one tackled in this thesis, in the sense that windows ‘arise’ from allowing the vehicle schedule to be modified, rather than existing in the original vehicle schedule, there is a direct association between the reasons for modifying a flight departure time and those for relieving inside a WRO in our problem. In principle, this formulation (and the algorithms proposed for it)

might be re-interpreted and reused for (T)DSW, by modeling the WROs in TDSW as time windows for vehicle departure times. However, it must be noted that the formulation is some ways more restrictive than TDSW, since given a choice of vehicle departure times, relieving is then restricted to those departure times only, whereas in our formulation the full WRO is in principle always available for relieving. However, in Chapter 4 we propose a method that loops over GaS and Local Search phases, where each GaS phase is run with a set of re-timed relief points within WROs derived from the previous local search phase; in this case the two underlying models are very similar.

Other authors also consider time windows in aircraft scheduling as a facilitator for increased efficiency within integrated planning of operations. Ahuja et al. [3] develop models for a combined through and fleet assignment problem with time windows. The authors claim that time windows during fleet assignment allow for greater opportunities between flight legs, which in turn allows for improvements on the combined through and fleet assignment problem. A multicommodity network flow model is first introduced, but the resulting problem is too large to be solved to a reasonable quality; hence, they develop a neighbourhood search algorithm instead, which they claim produce improvements over existing solutions.

A Tabu Search Approach for Driver Scheduling with WROs

To our knowledge, the problem of driver scheduling with windows of relief opportuni- ties is only considered in Shen’s PhD thesis [69] and a technical report by Shen and Kwan [71]. In her PhD thesis, Shen initially concentrates on solving the TDS prob- lem without consideration of WROs. A neighbourhood search scheme is proposed, which considers three possible ways (moves) to modify the existing schedule:

1. ‘swap links’: this move is similar to the 1-point crossover described in Section 4.2.1.

2. ‘replace AROs’: this move is similar to the transfer of pieces of work (at piece- level only) in Section 4.2.1

3. adding a shift: because their solution can be infeasible and not cover all vehicle work, a specific move to efficiently add a new duty to an existing solution is

proposed. Some work may be removed from existing shifts in the schedule when adding a new shift.

Based on the neighbourhoods described above, Shen then proposes a Tabu Search approach that is embedded into the algorithm described in Algorithm 1.

Algorithm 1 HACS (Shen and Kwan)

1: construct a (possibly infeasible) initial schedule, using a heuristic

2: minimize total penalty (using Tabu Search)

3: minimize total cost (using Tabu Search)

4: if current best schedule is feasible then

5: END

6: else if if total cost has been reduced then

7: go to step 2

8: else

9: use an extra shift to reduce penalty, go to 2

10: end if

Shen and Kwan then develop an extension of HACS for the TDSW problem. WROs are modelled using 1-minute expansions (Section 2.2.2). The ‘replace AROs’ move is modified to account for WROs. This is done by modifying the interpretation of a predecessor and successor of a RO in a vehicle block (where the order relation is given by the relief time of ROs) in the context of a 1-minute expansion of WROs. For example, given a vehicle block with one RO at location A at 12.00 and one (at- tended) WRO at location B between 12.30 and 12.31, the predecessor of (B, 12.30) is (A, 12.00), while the predecessor of (B, 12.31) is (B, 12.30). The ARO-replacing move is then redefined to look at predecessors and successors of AROs in terms of the new definition. This equates to considering the work between each pair of consecutive relief times within a WRO as a separate piece of block for the purposes of this move. New moves (swapping spells and inserting spells) are also added.

Experiments are carried in a set of 10 instances, mostly small in size (only one of the ten instances is comparable in size to those used in Chapter 5 in this thesis). Moreover, instances are simplified (e.g. by eliminating unattended WROs). It is not clear from the thesis or report which side constraints (e.g. labour agreement rules)

apply to the test instances. In most cases, the test instances are RoA formulations where WROs are added artificially, using a uniform 5-minute WRO length; our expe- rience in this thesis suggests that this may overestimate the amount of time windows available on real-life instances (for example, most WROs in the InterCity dataset are two or three minutes long; most WROs in the ThamesLink dataset are only one or two minutes long). On these instances, experiments on HACS starting from solutions not obtained by TrainTRACS are on average 0.55% higher cost than those obtained running TrainTRACS on a RoA formulation, while experiments on HACS starting from the solutions obtained with TrainTRACS over the RoA formulation show an average decrease in cost of around 1%.

Discussion The work by Shen and Kwan provides a good starting point to the study of WROs in the context of train driver scheduling. Some central concepts are introduced, like the idea of a 1-minute expansion of WROs, and that of retiming an active relief on a WRO, although the latter is only exploited in a very limited way, i.e. by moving the relief point one minute at a time in the context of ‘replace ARO’ moves.

In our view, the main limitations in this work are, first, that no study is con- ducted on how the problem of driver scheduling changes when WROs are introduced; second, the algorithms are developed for simplified versions of the problem, and WROs in the test instances are artificially generated –hence, it is difficult to trans- late the gains in performance shown in the experiments with actual gains achievable in a real-life context. At the same time, existing tools for the TDS problem are only exploited in a very limited way, e.g. to provide starting solutions. In this thesis we try to address these three perceived limitations to advance the study of the TDSW problem.