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W. Zhou, et al.: Crew Scheduling Considering both Crew Duty Time Difference and Cost on Urban Rail System WENLIANG ZHOU, Ph.D.1 Traffic Management (Corresponding author) Original Scientific Paper E-mail: [email protected] Submitted: May 22, 2015 XIA YANG, Ph.D.2 Accepted: Apr. 27, 2016 (Corresponding author) E-mail: [email protected] JIN QIN, Ph.D.1 E-mail: [email protected] LIANBO DENG, Ph.D.1 E-mail: [email protected] 1 School of Traffic and Transportation Engineering Central South University No.22 ShaoShan South Road, Changsha 410075, China 2 Department of Civil and Environmental Engineering Rensselaer Polytechnic Institute, Troy, NY 12180, USA CREW SCHEDULING CONSIDERING BOTH CREW DUTY TIME DIFFERENCE AND COST ON URBAN RAIL SYSTEM ABSTRACT expenses, and a small improvement to the crew sched- ule essentially leads to accumulated savings produc- Urban rail crew scheduling problem is to allocate train services to crews based on a given train timetable while sat- ing large annual cost savings for urban rail enterprise. isfying all the operational and contractual requirements. In Both the solving difficulty and the remarkable this paper, we present a new mathematical programming practical significance of URCSOP have attracted a model with the aim of minimizing both the related costs of lot of interest from many researchers. Morgado and crew duty and the variance of duty time spreads. In addition Martins [2] attempted to generate multiple alternative to iincorporating the commonly encountered crew sched- crew schedules based on different scheduling criteria uling constraints, it also takes into consideration the con- for the Portuguese rail. However, URCSOP was more straint of arranging crews having a meal in the specific meal frequently formulated as either set covering problem period of one day rather than after a minimum continual ser- or set partitioning problem with binary integral deci- vice time. The proposed model is solved by an ant colony al- gorithm which is built based on the construction of ant travel sion variables, and each decision variable represents network and the design of ant travel path choosing strategy. whether or not a duty is chosen as the service for a The performances of the model and the algorithm are evalu- crew. Both problems were solved by either an exact ated by conducting case study on Changsha urban rail. The or a heuristic method, or their combination. For exam- results indicate that the proposed method can obtain a sat- ple, Kroon and Fischetti [3] built a set covering model isfactory crew schedule for urban rails with a relatively small of crew scheduling for the Dutch railway, and solved it computational time. using dynamic column generation techniques, Lagran- gean relaxation and powerful heuristics respectively. KEY WORDS Alfieri et al. [4] also proposed a set covering problem urban railway; crew schedule; ant colony algorithm; duty based on an implicit column generation solution ap- time difference; proach for scheduling train crews on a rail network. Nishi et al. [5] proposed a column generation with dual 1. INTRODUCTION inequality for rail crew scheduling, and Jütte et al. [6] adopted a column generation based decomposition Urban rail crew schedule optimization problem algorithm to solve URCSOP which was decomposed (URCSOP) aims to find the lowest cost crew schedule into overlapping regions and optimized in parallel in covering all trips of trains specified in a train timetable this method. Column-generation-based methods have subject to various constraints such as crew rest and higher solving efficiency than other exact approaches, meal time requirements. It is difficult to solve URCSOP and they are widely used to solve the allocation prob- with a large number of train trips and complicated op- lems, but as stated in study [7], they are not efficient erational and contractual requirements. Fischetti et al. for solving URCSOP whose decision variables are only [1] have pointed out that the URCSOP is an NP-hard determined as 0 or 1, rather than any real number. problem. Nevertheless, the URCSOP has still become As enormous train trips should be serviced by one of the most important topics in urban rail trans- crews in URCSOP, the solving efficiencies of exact port because crew cost is the main variable operation methods are not satisfactory. They usually suit to Promet – Traffic&Transportation, Vol. 28, 2016, No. 5, 449-460 449 W. Zhou, et al.: Crew Scheduling Considering both Crew Duty Time Difference and Cost on Urban Rail System handle small to medium size problems. Therefore, The rest of this paper is organized as follows. First, some researchers tried to apply other types of ap- a brief description of URCSOP is presented, and then proaches to solve the large-scale real-word URCSOP the costs related to crew duty are analysed in Section with a large number of train trips and many types of 2. Next, URCSOP is modelled as a set partitioning constraints. For instance, Freling et al. [8] tried to use problem minimizing both crew costs and duty time dif- a decision support system to solve rail and airline crew ferences in Section 3. After that, a solving algorithm planning problems. Vaidyanathan et al. [9] proposed a based on the ACO algorithm is given in Section 4. Com- network flow-based approach to solve the URCSOP, in putational results of the Changsha urban railway on which crew duties satisfied the first-in-first-out rule on this proposed approach are introduced in Section 5. the network. And Jütte and Thonemann [10] proposed Finally, the conclusion and recommendations for fur- a graph partitioning strategy for the large-scale crew ther studies are provided in Section 6. scheduling problems. In addition, meta-heuristics has nowadays become 2. PROBLEM DESCRIPTION AND CREW a popular method because of its good performance in DUTY COST ANALYSIS solving the complicated, large-scale real-world prob- lems although they usually only find the near-optimal The urban rail consists of a set of crew home sta- solutions. Regarding their application in crew schedul- tions, a set of meal stations, a set of rest stations and ing problems, Emden-Weinert et al. [11] used a sim- a set of transition stations, which are denoted by Sh, ulated annealing approach to solve the airline crew Sm, Sr and St respectively. Each crew has to start and scheduling problem, and Dias et al. [12] proposed a end their duty at a home station, have a meal at a meal genetic algorithm for bus driver scheduling problem. station, and have a rest at a rest station. Moreover, Moreover, Elizondo et al. [13] presented a constructive each crew is only allowed to switch to another train for hybrid method to address the URCSOP, and provided continuing their duty at a transition station after they better results with regard to idle time than both the complete the current duty for one train. Generally, a hybrid and greedy methods. Hanafi and Kozan [14] station can simultaneously belong to multiple different applied a hybrid constructive heuristic with the simu- types of stations. For example, generally a meal or rest lated annealing search algorithm to solve the rail crew station is also a transition station. As crews are only scheduling problem. allowed to start and end their duties or change their Ant colony optimization (ACO) algorithm proved by service trains at the above four types of stations, we Dorigo [15, 16] to have higher efficiency in solving com- uniformly call them duty-division stations for descrip- binatorial optimization problems. Comparing with oth- tion purposes in this paper. er heuristic methods, it has some nice features such The whole trip of each train predetermined by the as parallel computing, high robustness and simplici- ty of operation. However, very few works in literature train timetable must be serviced by either a group of applied the ACO algorithm to solve the crew sched- crews or multiple groups of crews. A train trip can be uling problems. The aim of this paper is to apply an divided into a sequence of trip segments by the duty-di- ACO algorithm to solve the URCSOP considering both vision stations, and each trip segment can be serviced crew cost and duty time difference. Compared with the by different groups of crews. As illustrated in Figure 1 existing studies, our research has the following three in which seven types of lines are used to display trips main improvements. of different trains, the whole trip of the train represent- 1) Aim to minimize not only the related costs of crew ed by dotted lines is divided into two trip segments duties, but also the variance of duty time spreads, numbered 2 and 3 by the meal station. Each train trip which contributes to make crews of different duties segment has to correspond to a crew duty segment have nearly equal service time, and improves the which is the basic composition unit of crew duty and equity of crew service. can be viewed as crew minimum continuous service 2) Consider the requirement of crews being arranged unit. Therefore, crews only start and stop or change to have a meal in the specific meal periods of one their duties at both ends of each duty segment. day rather than after a minimum continuous ser- Generally, the URCSOP is to specify the service se- vice time. Obviously, the former is more in accor- quence of train trip segments to form a duty for each dance with the crews’ healthy eating habits. group of crews with the aim of minimizing the related 3) Design an ACO algorithm to solve our proposed costs of crew duty while satisfying urban rail operation- model for obtaining a higher solving efficiency al and contractual requirements.

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