# Branch And Bound Technique For Assignment Problem Hungarian

School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China

Copyright © 2016 Chao Lu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is concerned with the scheduling of Electrical Multiple Units (EMUs) under the condition of their utilization on one sector or within several interacting sectors. Based on the introduction of the train connection graph which describes the possible connection relationship between trains, the integer programming model of EMU circulation planning is constructed. In order to analyzing the resolution of the model, a heuristic which shares the characteristics with the existing methods is introduced first. This method consists of two stages: one is a greedy strategy to construct a feasible circulation plan fragment, and another is to apply a stochastic disturbance to it to generate a whole feasible solution or get a new feasible solution. Then, an exact branch and bound method which is based on graph designing is proposed. Due to the complexity, the lower bound is computed through a polynomial approximation algorithm which is a modification from the one solving the degree constraint minimum 1-tree problem. Then, a branching strategy is designed to cope with the maintenance constraints. Finally, we report extensive computational results on a railway corridor in which the sectors possess the basic feature of railway networks.

#### 1. Introduction

The assignment of transportation tools to fulfilling a group of tasks under certain conditions is one of the most important scheduling activities in many real world applications, such as the Aircraft Routing Problem (ARP) in airline operation, Vehicle Routing Problem (VRP) in logistics delivery, and the Locomotives Scheduling Problem (LSP) in railway system. As the mobile resources, the quality of the utilization of the transportation tools has a great effect on the efficiency and the economic benefits of the transportation enterprises. The operation of these resources also provides a foundation for the conduction of other works and has an interaction with them.

In high-speed railway network, Electrical Multiple Units (EMU) are the main tools used for the passenger transportation. They are significantly cost equipment, because not only is their acquisition or construction expensive but also they need power supply and regular maintenance. The scheduling of EMU falls in the category of rolling stock circulation problems and the operation plan of them determines the concrete activities they need to perform when the system is running. In order not to result in the waste of the precious resources, they need to be operated as reasonably as possible.

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. 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. Among these plans, the EMU circulation plan mainly determines the connection relationship of the train trips and arranges the first level maintenance for EMUs. It is a basis for the other two plans.

The optimization of the rolling stock circulation scheduling problem has received a lot of attention from researchers. Some realistic problems such as the unit coupling or uncoupling when scheduling the EMU are also considered. Cacchiani et al. [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. Cacchiani et al. [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. Abbink et al. [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. Peeters and Kroon [5] focused on the efficient circulation of train units within a certain scope of railway line given the timetable and the passengers’ seat demand and a branch-and-price algorithm is described. Alfieri et al. [6] present a solution approach based on an integer multicommodity flow model to determine the appropriate numbers of train units of different types together with their efficient circulation on a single line. Cacchiani et al. [7] present two integer linear programming (ILP) formulations together with their relaxations (the linear programming (LP) relaxation and Lagrangian based approach, resp.) to assign the train units to the trips with minimum cost. When the high-speed railway system is running, many disturbances can occur and that would lead to the irregularities of the operations of the train units. In order to avoid such events to some extent, it is reasonable to find a plan that is insensitive to these disruptions; that is, the plan is able to cope with relatively small disruptions without structural changes, which is also known as the robustness scheduling, or to react immediately to those disruptions by applying a recovery strategy to the plan defined previously, which is also known as the rescheduling process (see, e.g., [8–10]). Cadarso and Marín [11] formulate a multicommodity flow model for the rolling stock problem in rapid transit networks. 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 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. [13]). 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. Locomotives are often scheduled on a fixed sector and could be viewed as an assignment problem which could be solved by Hungarian algorithm. Ahuja et al. [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. 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. On the basis of that, Miao et al. [16] transformed the original problem into a multiple Traveling Salesperson Problem with replenishment and designed a hierarchical optimization heuristic algorithm. Shi et al. [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. Li et al. [18] introduces the optimized EMU connection graph, based on which the improved particle swarm optimization algorithm is designed for solving the problem.

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. The algorithm is based on the graph theory and could be able to deal with the problems of practical size within a reasonable time. Furthermore, we propose a heuristic, which shares the characteristics with the existing methods and is based on local search strategy. We also make a comparison between these methods.

The paper is organized as follows. In Section 2, we describe the details of the EMU circulation scheduling problem we study. 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. In Section 4, we design the neighbor structure of the solution of the problem and the local search method is illustrated. In Section 5, the exact branch and bound algorithm for solving the problem is outlined, and the details of the algorithm such as the calculation of the approximate lower bound and the branching strategy is illustrated. In Section 6, we report the computational results of the comparison of the proposed methods in a test case.

#### 2. Problem Description

In high-speed railway network, EMUs are utilized on one railway sector or within several interacting sectors; see Figure 1. There are a set of train trips of up and down direction in each sector. The EMUs circulation scheduling problem needs to assign all the tasks of the train trips to a set of EMUs. In each sector, the train trips of the up direction and the train trips of the down direction are connected at the endpoint stations by certain EMUs. At the stations which are linked with the EMU maintenance base, the EMUs need to have maintenance if either the accumulative running distance or running time reaches the upper bound which is provided in the first level maintenance document of the EMU.

**Figure 1: **Utilization of EMU within several interacting sectors.

Given a train timetable, the circulation plan could be shown as the format in Figure 2. During a period of one day, we use the term circulation plan fragment to indicate a series of tasks that an EMU should fulfill, which includes several train trips and train connections at stations. In Figure 2, a circulation plan fragment is represented by a polygonal line that links the tasks within one day. Each two train trips could be connected at the station if the duration between the arrival time of one train trip and the departure time of another train trip is not less than the provided minimum duration time. If the EMU needs to have maintenance, then the duration of the connection should also exceed the minimum duration of the maintenance. For example, if an EMU conducts the tasks which are provided in fragment ① of the circulation plan in the first day, it starts from station A and finally arrives at station B after all the tasks have been conducted; it stay at station B in the night. In the second day it starts from station B and conducts the tasks which are provided in fragment ② of the circulation plan. After it finishes serving as train 3102, it has maintenance at station A. Finally, after conducting all the tasks it returns to station A and stays there in the night. In the following days it should conduct the tasks which are provided in another fragment of the circulation plan.

**Figure 2: **Fragments of the circulation plan.

The whole circulation plan is formed by connecting all the fragments through the overnight connection When the first EMU conducts the fragment ① of the circulation plan, other EMUs conduct the fragments ②, ③, ④ of the circulation plan according to the same rules respectively, then the number of the utilized EMU is known. The maintenance in the circulation is of the basic level and the extensive maintenance which needs much longer time, therefore they are not considered in the circulation plan.

It is worth mentioning that one of the objectives pursued in the scheduling process of the rolling stock circulation would consider maximization of service to the passengers by minimizing the seat shortages. However, under the Chinese railway organization mode the types of trains are viewed as the same and the number of travelers of each sector is dealt with when working out the train operation scheme which provides the train service frequency. Therefore, the consideration of the number of travelers when solving the EMU circulation scheduling problem is out of the scope of this paper.

#### 3. Model of the EMU Circulation Planning

##### 3.1. Train Connection Graph

Given a timetable of several interacting railway sectors, the corresponding EMU connection graph could be constructed correspondingly, which is a useful tool to describe the essentials of the problem of EMU circulation scheduling. The notations with regard to the problem are listed in Notations section.

The parameter if and only if there could exist a connection relationship between trains and under the condition of and . is calculated as follows:

Parameter if and only if and .

Figure 3 shows the connection graph of a certain group of routes, in which we only display the arcs from the up-bound trains to the down-bound trains. Each row of nodes represents the set of trains that belong to the same route. The solid arcs are the connection arc whose duration time is within one period, while the dotted arcs are the connection arc in which the departing train and the arriving train is distributed in two periods.

**Figure 3: **The train connection graph.

##### 3.2. Model Formulation

We use the binary decision variables and to indicate whether the arc in the connection graph is selected in the circulation plan and used as the maintenance arc, respectively, that is, if and only if train is scheduled to be connected by train in the circulation plan; then, . if and only if the arc is used as the maintenance arc in the circulation plan. Then, the model of EMU circulation planning could be formulated as follows:

Objective function (2) consists of two terms, the first term is the total connection time, which is closely related to the number of EMUs needed and this is the main objective of this problem, the second term is the total times of the maintenance. Constraints (3) and (4) impose that each train should be used as the starting node and the ending node of only one arc in ; that is, each node of train trip should be visited only once in the circulation plan. Constraints (5) impose that the connection between trains should be reasonable. Constraints (6) and (7) impose that the cumulative running time and running distances of an EMU between two maintenance arcs must not exceed the upper bound of the prescriptive distance and time, where is the set of the maintenance arcs. The set contains all the trains between maintenance arcs and . Constraints (8) impose that an arc could be selected as the maintenance arc if and only if that arc is used as a connection arc and is able to provide the maintenance. Constraints (9) avoid the occurrence of the subloop. Given a certain arc set connecting all the nodes in (see Figure 4), when arbitrarily deleting two arcs (represented by the hidden line), then with the starting or ending node of the deleted arc as the root node and searching all the connected nodes using the adjacency matrix of the existing arcs the number of the nodes that can be found is denoted as , which is linked by the dotted line in Figure 4. Obviously, (a) is subjected to constraints (9) while (b) is not subjected to constraints (9). Constraints (10) avoid the existence of none maintenance loop for the circulation plan.

**Figure 4: **Illustration of the notation .

#### 4. Heuristic

##### 4.1. Generation of the Initial Solution

Through the analysis of the constraints of the model, the format of the solution could be written as

This paper presents a new branch-and-bound algorithm for solving the quadratic assignment problem (QAP). The algorithm is based on a dual procedure (DP) similar to the Hungarian method for solving the linear assignment problem. Our DP solves the QAP in certain cases, i.e., for some small problems (*N*< 7) and for numerous larger problems (7≤ *N* ≤16) that arise as sub-problems of a larger QAP such as the Nugent 20. The DP, however, does not guarantee a solution. It is used in our algorithm to calculate lower bounds on solutions to the QAP. As a result of a number of recently developed improvements, the DP produces lower bounds that are as tight as any which might be useful in a branch-and-bound algorithm. These are produced relatively cheaply, especially on larger problems. Experimental results show that the computational complexity of our algorithm is lower than known methods, and that its actual runtime is significantly shorter than the best known algorithms for QAPLIB test instances of size 16 through 22. Our method has the potential for being improved and therefore can be expected to aid in solving even larger problems.

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