Discrete OptimizationA decompose-and-fix heuristic based on multi-commodity flow models for driver rostering with days-off pattern
Introduction
Public urban transport companies face an increasing burden regarding operating costs, being expenditure on staff the bulk of these costs. Efficient management of human resources is needed, in particular, regarding staff engaged to the driving of buses. Therefore, companies require the enhancing of techniques for planning bus drivers’ work.
The driver rostering problem in public transit companies aims at assigning daily crew duties to each driver defining a sequence of workdays and days-off which is the driver schedule to be in force during a rostering period of pre-determined length, the rostering horizon. A roster is the set of all driver schedules, together with the particular work shifts that drivers must work on. The rosters must comply with law, labor regulations and drivers’ union agreements, and also meet the demand for transport in specific urban areas. Such a problem is usually highly constrained and dependent on the particular contexts where it may arise.
We can step back until at least the sixties of the last century to find Operational Research studies on bus drivers scheduling and rostering (e.g. Bennett & Potts, 1968). Since then, a variety of surveys may be found, either on edited volumes of conferences or on specific papers that present methods and/or models for dealing with problems in specific cases, as surveyed by Ernst et al., 2004a, Ernst, Jiang, Krishnamoorty and Sier, 2004b. More recent publications include the survey by Van den Bergh, Beliën, De Bruecker, Demeulemeester, and De Boeck (2013), a reference model for timetabling and rostering proposed by Causmaecker and Berghe (2012), as well as, other works on rostering in different transport contexts. Lusby, Dohn, Range, and Larsen (2012) propose a column generation based heuristic to roster ground staff with days-off pattern for an airline company and Nishi, Sugiyama, and Inuiguchi (2014) solve a railway rostering problem in a cyclic context by a Benders decomposition algorithm. Xie, Kliewer, and Suhl (2012) integrate the cyclic crew scheduling and driver rostering problems and propose a multiobjective network problem solved with standard software followed by a simulating annealing for improvement on one of the objectives. Other relevant study for crew rostering in a cyclic environment is Hartog, Huisman, Abbink, and Kroon (2009) where a planning process uses integer linear programming models to build train driver rosters. Nurmi, Kyngäs, and Post (2011) propose a model that separately schedules days-off and shifts within a non-cyclic environment. The problem is solved with a population-based local search heuristic that is applied in a Finish bus transit company. Other studies involving population based heuristics (evolutionary or scatter search) are presented in Respício, Moz, and Pato (2013) for non-cyclic bus driver rostering, Burke et al. (2010) and Maenhout and Vanhoucke (2010) for airline rostering and Dornberger, Frey, and Hanne (2008) for railway rostering.
Multi-commodity multilayer network flow models have been widely used to describe rostering problems, e.g. Aronson (1986) and Aringhieri and Cordone (2004) for personnel rostering in general and Moz and Pato (2003) for nurses. As to the authors’ knowledge, the first formulation for bus driver rostering in a multilayer network is due to Carraresi and Gallo (1984). In transport contexts, several studies involving multi-commodity flow models were published, namely, the ones by Cappanera and Gallo (2004) for airline crew rostering and Xie and Suhl (2014) for cyclic and non-cyclic crew rostering problems in public bus transit.
The authors studied the bus driver rostering as a sub-problem of the integrated vehicle-crew-roster problem. Firstly, in Mesquita et al. (2011), a roster problem with no pre-defined pattern was addressed in the last step of a sequential algorithm. Later on, in Mesquita, Moz, Paias, and Pato (2013), a roster problem where drivers work according to a pre-defined days-off pattern was studied as a sub-problem of a Benders decomposition approach. Although good quality results have been attained for the integrated vehicle-crew-roster problem with days-off pattern, the rostering part revealed to be too CPU time consuming which pointed to deeply research on the rostering problem on its own.
In what follows, Section 2 presents the description of the bus driver rostering problem with days-off pattern (DRP). Section 3 is devoted to the theoretical complexity of the problem. In Section 4 three mixed integer linear programming formulations (MILP) are presented: an assignment/cover model, a multi-commodity flow model and a new multi-commodity flow/assignment model, the last two defined on multilayer networks. In Section 5 the linear relaxations of the models are studied with respect to the bounds they provide for the optimum value of DRP. Section 6 describes the new decompose-and-fix heuristic approach. Computational experience, carried out on a set of instances derived from real world and benchmark data, will be presented and commented in Section 7. Lastly, some conclusions will be drawn in Section 8.
Section snippets
Problem description
In most public urban transport companies, drivers are divided into small groups depending on several factors like seniority, employment relationship with the company, and urban area to cover. The rostering for each group may stand on different strategies.
The problem here presented is devoted to the assignment of a set L of crew duties to a pool of |M| drivers who work according to a pre-defined days-off pattern without cyclic crew duty assignments, during a rostering horizon of |H| days.
The set
Theoretical complexity
The DRP is NP-hard. This may be proved by showing that there is a known NP-hard problem, the set covering problem with equal objective function coefficients, which is polynomially transformable to the DRP (Nemhauser & Wolsey, 1988). In the sequence, we will see that the DRP with cyclic days-off pattern is also NP-hard.
Proposition 1 The DRP is NP-hard. Proof Let us consider instances of the rostering problem such that:
; rm = 1, ∀m ∈ M; there are only normal early crew duties, thus there are neither
Mathematical formulations
This section presents three mathematical formulations for DRP. It begins by a brief presentation of an assignment/covering model that directly uses the rostering matrix. Afterwards, to take advantage from network characteristics of the rostering problem, two multi-commodity network flow formulations are developed.
In all formulations, the objective function measures the quality of the solution in terms of driver costs, related to the drivers assigned to work and the respective salaries, as well
Linear relaxation bounds
In this section the lower bounds obtained from the linear programming (LP) relaxation of models AC, MF and MFA – – are compared from a theoretical point of view.
Proposition 3 The optimal value of model, , is less than or equal to the optimal value of model, , and, for some instances, it can be strictly lower. Proof The relaxation of the integrality constraints (MF6), (AC8) and (AC9) follows:
Decompose-and-fix heuristic for MF/MFA models
All mathematical models proposed in this study involve a large number of integer variables and a significant dual degenerescence. Consequently, the use of a generic MILP solver will consume a great amount of computing resources often stopping out of memory without finding any feasible solution. Therefore, heuristic methods are developed by taking into account the structure of the mathematical models and the potentialities of the MILP solver. MILP-based heuristics have been developed for
Computational experience
The three different models were tested with instances derived from real world data and from benchmark data. More precisely, we have compared the LP relaxation solutions and also the feasible solutions obtained by the heuristics. All the algorithms were coded in C using the CPLEX Optimization Studio Academic Research 12.5 library and ran on an Intel® Core™ i7-2620M CPU 2.70 gigahertz RAM 8.00 gigahertz. The time limit for the branch-and-bound was set to 10,800 seconds.
Conclusions
This paper proposes two multi-commodity flow models, MF and MFA, for the driver rostering problem with days-off patterns and without cyclic crew duty assignments. Theoretical properties of the problem and of the models have been studied. The underlying network structure of MF and MFA suggested a decompose-and-fix heuristic to solve the rostering problem. The basic idea of this heuristic is to identify in a mathematical formulation a set of sub-problems to be solved according to a hierarchic
Acknowledgments
This research was supported by Portuguese National Funding from FCT under project PEst-OE/MAT/UI0152. The authors thank the company Carris de Ferro de Lisboa, Portugal, for the helpful discussions.
The authors would like to thank the anonymous referees for the helpful comments.
References (28)
- et al.
The multicommodity multilevel bottleneck assignment problem
Electronic Notes in Discrete Mathematics
(2004) The multiperiod assignment problem: A multicommodity network flow model and specialized branch and bound algorithm
European Journal of Operational Research
(1986)- et al.
A multi-objective approach for robust airline scheduling
Computers & Operations Research
(2010) - et al.
A multi-level bottleneck assignment approach to the bus driver's rostering problem
European Journal of Operational Research
(1984) - et al.
Staff scheduling and rostering: A review of applications, methods and models
European Journal of Operational Research
(2004) - et al.
A hybrid scatter search heuristic for personalized crew rostering in the airline industry
European Journal of Operational Research
(2010) - et al.
A decomposition approach for the integrated vehicle-crew-roster problem with days-off pattern
European Journal of Operational Research
(2013) - et al.
Set partitioning/covering-based approaches for the integrated vehicle and crew scheduling problem
Computers & Operations Research
(2008) - et al.
Personnel scheduling: A literature review
European Journal of Operational Research
(2013) A guaranteed-accuracy round-off algorithm for cyclic scheduling and set covering
Operations Research
(1981)
Rotating roster for a transit system
Transportation Science
A multicommodity flow approach to the crew rostering problem
Operations Research
Towards a reference model for timetabling and rostering
Annals of Operations Research
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