Although there is no agreed consensus on the exact definition of a metaheuristic, we will take Blum and Roli’s view [7], who say that a metaheuristic is
a new kind of approximate algorithm [. . . ] which tries to combine basic heuristic methods in higher level frameworks aimed at efficiently and effectively exploring a search space.
In the last twenty years or so research in metaheuristics and its application to scheduling problems has become very popular. Most of the major metaheuristics have been (and are still) adapted to and tested on the crew scheduling or rostering problem for either either airline, bus or train operations.
We present here a brief overview of how different metaheuristics have been ap- plied to crew scheduling. We concentrate on the choice of solution representation, where a very wide range of proposals is exhibited. This is undoubtedly linked to the fact that metaheuristics impose very few constraints on the underlying repre- sentation (as opposed to e.g. linear programming). However, it must be noted that
some questions are only relevant to particular metaheuristics – for example, genetic algorithms seem to be much more sensitive to solution representation than other metaheuristics, in particular when the standard mutation and crossover operators are used. Where relevant, we also highlight how metaheuristics are modified to incorporate elements that are not part of the standard formulation. We refer the reader to Section 3.3 for a more comprehensive look at the key metaheuristics in the context of their potential application to problem of driver scheduling with WROs. Tabu Search Cavique et al. [12] propose algorithms for a decision-support system for crew scheduling at Lisbon Underground. Initial solutions are built with a well- known problem-specific heuristic. In their first Tabu Search proposal, the next solution in the search is built by removing inefficient shifts from the current solution and replacing them with more efficient shifts (which the authors claim to be a form of ‘strategic oscillation’). In their second proposal, moves do not operate on the solution space directly, but rather indirectly on the space of possible partitions of the set of vehicle blocks into pieces. A matching graph G is built over this partition, and finally a schedule is obtained by solving a maximum-cardinality matching problem over G. Moves modifying the partition are implemented as sequences of simple moves; these sequences are usually referred to as ‘subgraph ejection chains’ in Tabu Search literature. Shen and Kwan [70, 69] propose a tabu search approach for the TDSW problem; their approach is described in more detail in Section 3.2.1.
Simulated Annealing Emden-Weinert and Proksch [24] present a simulated an- nealing approach for airline crew pairing. Their application of simulated annealing is quite straightforward, although they introduce an extension where after the com- pletion of each temperature level a deterministic local improvement phase is fired, exploring specific compound moves.
Evolutionary Computation Levine [58] presents a steady-state genetic algo- rithm (GA) for airline crew scheduling, where in each iteration the worst-ranked solutions (usually one or two) in the current set are replaced with new ones. A chromosome contains one bit per ‘column’ (i.e. candidate shift) in the set-covering formulation (each bit can then be directly associated with a 0-1 variable in the
set-covering formulation). Recently, Kotecha et al. [52] claim to have improved on the results from Levine on another GA proposal which uses a different, row-based encoding, and a cost-biased crossover.
Kwan [53] and Kwan et al. [57] study variants of GA where so-called ‘combi- natorial traits’ (features of a solution that are deemed positive) are detected on elite solutions, and are carried forward to offsprings. Initial solutions are built with information derived from solving the LP relaxation of a set covering model, which provides with an initial list L of preferred shifts. A chromosome does not represent a full solution; instead, it contains one bit per preferred shift in L. Their chromosome representation and crossover/mutation operators mean that in most cases the chro- mosomes will not describe a valid schedule; hence, a greedy repair heuristic stage is added. The repair heuristic presented in this work predates those introduced in our proposal in Chapter 6.
Current Research Research in metaheuristics for transportation scheduling is still very active. As we discuss in more detail in Section 3.3, there is currently a research trend in hybridizing search techniques in order to solve a combinatorial opti- mization problem. Among the presentations on the most recent CASPT conference (2006), Moz et al. [64] propose two evolutionary (meta)heuristics for a bi-objective formulation of the bus driver rostering problem, one of which integrates local search into an evolutionary framework. Similarly, Souai and Teghem [72] propose a hy- bridization of genetic algorithms and simulated annealing, where the latter is used to restore feasibility to the solutions generated by the genetic algorithm under their crossover and mutation operators.
Another promising development that can be linked to the area of metaheuris- tics is the so-called ‘hyper-heuristic’ approach introduced by Soubeiga et al. [13]. The key idea of hyper-heuristics is to provide a high-level, problem-independent framework to control the execution of lower-level, presumably very simple problem- specific heuristics. A key suggestion is the use of a ‘choice function’ that dynamically ranks the low-level heuristics during the search, ideally using only minimal informa- tion from the heuristics, such as improvements generated by each call to a specific heuristic. The approach has been applied to the bus driver scheduling problem
by Rattadilok and Kwan [67], using an extension of the choice-function approach where the controller dynamically selects a set of parameters to instantiate a low-level heuristic to be executed during the search.