*It is a comprehensive plan and could be generally divided into three parts, namely, the EMU circulation plan, the EMU allocation plan, and EMU maintenance plan.*

The method is based on an approach in which sequences are designed once the timetable and the rolling stock assignment have been done.

As for the particular EMU circulation plan in China, it is mainly based on a condition of the utilization of the EMU on one railway sector or within several interacting sectors, which is generally viewed as a more reasonable mode for the EMU operation (see, e.g., Zhao et al. Although the scheduling of the EMU shares something in common with the scheduling of other kinds of transportation tools, especially the scheduling of locomotive in ordinary railway, it has its own characteristics.

However, the EMUs are scheduled within several linked sectors and some complicated constraints such as the EMU maintenance constraints must be taken into account; therefore, more applicable methods should be developed for solving the construction of the EMU circulation plan.

Zhao and Tomii [15] transformed the original problem into the Traveling Salesperson Problem and introduced a probability based local search algorithm, whose key points are about the connection of the trains and the generation of the maintenance arcs. [16] transformed the original problem into a multiple Traveling Salesperson Problem with replenishment and designed a hierarchical optimization heuristic algorithm. [17] designed a simulated annealing algorithm by introducing the penalty function and 3-opt neighborhood structure, which is based on the circular permutation of all trains. [18] introduces the optimized EMU connection graph, based on which the improved particle swarm optimization algorithm is designed for solving the problem.

Empty movements and shunting operations are considered and the robustness is introduced by selectively avoiding empty train movements and these operations.

Cadarso and Marín [12] presents a model to study the robust determining of the best sequence for each rolling stock in the train network.The algorithm is based on the graph theory and could be able to deal with the problems of practical size within a reasonable time. In Section 2, we describe the details of the EMU circulation scheduling problem we study.Furthermore, we propose a heuristic, which shares the characteristics with the existing methods and is based on local search strategy. In Section 3, we introduce the concept of the train connection graph, based on which the model of EMU circulation scheduling in high-speed railway network is constructed.Locomotives are often scheduled on a fixed sector and could be viewed as an assignment problem which could be solved by Hungarian algorithm. [14] formulated the locomotive-scheduling problem as a multicommodity flow problem with side constraints on a weekly space-time network.Each locomotive type defines a commodity in the network.The EMU operation plan in each country differs in small points.In China, the EMU operation plan defines the arranging of the train trips and maintenance tasks of each level to each EMU within a certain period of time according to the given train timetable, the configuration of EMU, the condition of EMU maintenance facilities, and the rules of the operation of the stations.Through the analysis of the studies concerning EMU circulation plan, it can be seen that the problem could be transformed into some classic optimization problems, and due to the complexity of these problems, most of the existing solution generation methods belong to the range of probability based searching heuristics.The motivation of this paper is to propose an exact algorithm for solving the EMU circulation scheduling problem in high-speed railway network.Some realistic problems such as the unit coupling or uncoupling when scheduling the EMU are also considered. [1] designed an effective heuristic procedure for the train unit assignment problem that were able to find solutions significantly better than the “manual” solutions found by practitioners. [2] proposed a heuristic which is based on the natural Lagrangian relaxation of a natural integer linear programming of the train unit assignment problem which turns out to be much faster in practice and still providing solutions of good quality. [3] deal with the tactical problem of finding the most effective allocation of the train types, subtypes, and units of rolling stock to the train series, such that as many people as possible can be transported with a seat, especially during the rush hours.Lin and Kwan [4] proposed a two-phase approach for the train unit scheduling problem; the first phase assigns and sequences train trips to train units considering some real-world scenarios and the second phase focuses on satisfying the remaining station detailed requirements.

## Comments Branch And Bound Assignment Problem

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