A hybrid constructive heuristic and simulated annealing for railway crew scheduling

Highlights

• A mathematical model and algorithms are developed for railway crew scheduling.

• The number of crew duties is minimised by reducing the total idle transition times.

• The interval of relief opportunities is included into the model and algorithms.

• A hybrid constructive heuristic and SA metaheuristic is applied to solve the problem.

• Near-optimal crew schedules are obtained within an acceptable cpu time.

Abstract

Railway crew scheduling problem is the process of allocating train services to the crew duties based on the published train timetable while satisfying operational and contractual requirements. The problem is restricted by many constraints and it belongs to the class of NP-hard. In this paper, we develop a mathematical model for railway crew scheduling with the aim of minimising the number of crew duties by reducing idle transition times. Duties are generated by arranging scheduled trips over a set of duties and sequentially ordering the set of trips within each of duties. The optimisation model includes the time period of relief opportunities within which a train crew can be relieved at any relief point. Existing models and algorithms usually only consider relieving a crew at the beginning of the interval of relief opportunities which may be impractical. This model involves a large number of decision variables and constraints, and therefore a hybrid constructive heuristic with the simulated annealing search algorithm is applied to yield an optimal or near-optimal schedule. The performance of the proposed algorithms is evaluated by applying computational experiments on randomly generated test instances. The results show that the proposed approaches obtain near-optimal solutions in a reasonable computational time for large-sized problems.

Introduction

The crew scheduling problem (CSP) in the transportation industry represents a computationally difficult combinatorial optimisation problem. The large number of tasks (trips) to include and the complicated operational and contractual requirements are the main reason for the complexity of the problem. Nevertheless, the CSP has been one of the most important focuses of the transportation industry because it affects the company’s profitability and its service quality. An optimal crew schedule is essential to ensure efficient and reliable operations of transportation services. Furthermore, the cyclic nature of the crew scheduling application makes the CSP a good candidate for optimisation. A small improvement to the crew schedules can lead to accumulated savings to produce large annual cost savings. The difficulty of solving CSP yet its enormous practical significance, have led to a large number of proposed solution techniques. However, unlike the airline CSP which has been intensively studied, the railway crew scheduling is less cited in literature (Goumopoulos and Housos, 2004). Railway crew scheduling is domain specific and there has been no developed solving method has yet to be applied universally. Models and algorithms are designed mainly for a specific case and may not readily be applied in different applications.

Railway CSP aims at finding a minimum cost crew schedule. The schedule should cover scheduled trips in a train timetable subject to various constraints. Morgado and Martins (1992) presented early work on the crew scheduling application for the Portuguese railways. The system provides a possibility of generating alternative schedules using different scheduling criteria and enables evaluation of the cost of the solution considering a set of produced statistics. CSP is more frequently formulated mathematically, as either set covering problem or set partitioning problem, and then solved by exact solution approaches and heuristics (see for example in Caprara et al., 1997, Kroon and Fischetti, 2001, Freling et al., 2004). In both the set covering and the set partitioning formulations, the decision variable is a binary integer variable that represents whether or not a duty (roundtrip, pairing) is selected as work for a crew member. The constraint in the set covering problem consists of a matrix of binary values, which defines that each piece of work is covered by a duty at least once. Each column represents one possible pairing or work to be performed by an individual crew member over a defined period of time. The set partitioning problem is similar to the set covering problem, except that in the set partitioning formulation the constraint becomes equal to one, meaning that each task is covered exactly once. Alfieri et al. (2007) proposed a set covering problem based on an implicit column generation solution approach for scheduling train drivers on a railway sub-network. Feasible duties are constructed from a set of trips to be serviced by a number of train drivers, with the aim of minimising the number of duties and maximising the robustness of the schedule. A heuristic procedure is applied to obtain an initial feasible solution together with a heuristic branch-and-price algorithm based on a dynamic programming algorithm for the pricing-out of columns. The main difficulty in applying the exact methods to the CSP is that in determining all possible solutions. For the CSP with a large number of trips, there can be an unmanageably large number of possible roundtrips. As a consequence, the problem becomes a time-consuming process of enumerating all the possible roundtrips. Bangert (2012) noted that the method of enumeration is not realistic when the number of options is too large and cannot be practically listed.

Generally, CSP involves a large number of decision variables and it is restricted by many constraints. Fischetti et al., 1987, Fischetti et al., 1989 have shown that the bus crew scheduling belongs to the class of NP-hard problems. For this reason, there is a requirement of large-scale solution techniques such as column generation. The concept of column generation is to solve a sequence of reduced problems (master problem) in which each reduced problem contains a small fraction of the set of variables (columns). The sub-problem or an auxiliary problem is commonly formulated as a restricted shortest path problem. The restricted shortest path problem however, is difficult to solve and it also needs other optimisation schemes such as dynamic programming algorithms or branch-and-bound methods. Bengtsson et al. (2007) formulated a general crew pairing problem with the objective function being to minimise the cost of selected pairing and the cost of violating soft constraints. The research combines resource constraints, k-shortest path enumeration, and label merging techniques and shows that a column generation approach is able to heuristically solve large and highly complex railway pairing problems in a reasonable time. Given the size and complexity of the railway operation, the researchers indicate the necessity of combined optimisation techniques. Nishi et al. (2011) proposed a column generation with dual inequality for railway crew scheduling. Computational results have shown that the proposed technique can accelerate the convergence of conventional column generation for a large data set application. Yan and Tu (2002), however, stated that column generation based methods could be inefficient because when the crew scheduling is formulated as a traditional set covering problem, the obtained optimal solutions could be non-integer solutions. Other techniques should then be incorporated to refine the non-integer solutions.

De Leone et al. (2011) proposed a mathematical model to solve a CSP. Since their proposed model can only handle small- to medium-sized problems, a greedy randomised adaptive search procedure has then been offered to solve large instances. Network flow approach has also been used in several researches on CSP. An attempt towards this approach was proposed by Vaidyanathan et al. (2007). They describe a network flow-based approach to solve a CSP arising in North American railroads. The CSP is formulated as an integer program on a space–time network enforcing the first-in-first-out requirement by including side constraints with the objective of minimising several components of crew expenses. Due to the difficulty of applying the network flow approach to highly complex constraints, this method may be suitable only for small- to moderately-sized real-world problems.

Metaheuristics have become a popular approach in tackling the complexity of practical optimisation problems. Although metaheuristics cannot guarantee optimality of their solutions, they have shown a very good performance in solving real-world optimisation problems. Metaheuristics represent a general type of solution method that illustrates the interaction between local improvement procedures and higher level strategies to facilitate the algorithm for both escaping local optima and exhaustively searching a feasible region. Elizondo et al. (2010) proposed a constructive hybrid approach to address operation management problems that emerge in underground passenger transport. The results are compared with two alternative methods based on tabu search and a greedy heuristic. The tabu search technique provides better results with regard to idle time than both the hybrid and the greedy methods. Dias et al. (2002) proposed a genetic algorithm for bus driver scheduling, which is developed by using a new coding scheme and considering a complex objective function.

Simulated annealing (SA)-based algorithms have been noticed to produce good solutions to several combinatorial optimisation problems. Emden-Weinert and Proksch (1999) solved an airline CSP using a SA approach. The results show that the SA yields good quality solutions but requires longer processing times than simpler heuristics. Lučic and Teodorovic (1999) applied a SA approach to solve a multi-objective crew scheduling for an airline. In spite of the potential application of SA algorithms to solve combinatorial optimisation problems, there has been few crew scheduling related applications in the literature using the SA algorithm.

This paper presents a new mathematical model and a hybrid constructive SA (HCSA) algorithm to solve railway CSP. The mathematical programming model incorporates commonly encountered real-life railway crew scheduling constraints, particularly the integration of the interval of relief opportunities. To the best of our knowledge, the inclusion of the interval of relief opportunities into models and algorithms has not been studied in depth. The rest of this paper is organised as follows. In Section 2, a brief description of the problem is presented. In Section 3, solution techniques that include formulation of the mathematical programming model and the details of the proposed HCSA algorithm are given. Results of the computational experiments on each approach are provided in Section 4. Finally, Section 5 provides the conclusion and recommendations for further study.