Allocation method for transit lines considering the user equilibrium for operators☆
Introduction
In recent years, reform in the transit sector takes place in many countries, including changing the operation of transit from monopoly to sharing. Government monopoly of transit lines is no longer a favorable strategy due to poor service level and low efficiency caused by monopoly. Compared with monopoly, operation-sharing of transit lines could introduce competition to maintain the service level and efficiency. Demsetz (1968) mentioned that competition of the market is a way to enhance efficiency. Opening the market to competition has been necessary to create competitiveness in the transit sector. If transit lines are operated by multiple operators, competitive pressure will be induced among operators. Thus, operators will improve service levels and reduce operating costs to increase their competitiveness. Many countries allocating transit lines by introducing competition have so far been successful (Preston, 2005, Wu et al., 2012, Zhao and Huang, 2016).
However, there are some problems caused by introducing transit competition in operation-sharing. Each operator’s individual goal is to maximize his own benefit. Hence, operators may adjust operational strategies of their lines to gain more benefits after the authority allocating lines (Zhou et al., 2005). In general, the main operational strategy is to adjust the departure frequency of each transit line. When an operator is allocated a line to run, competition exists with other operators running on lines “attractive” to the same passengers. On the competitive line, operators tend to increase the departure frequency to compete for passengers, such as deploy more vehicles on these competitive lines. And an equilibrium state is obtained when no operator can increase his profit by changing his departure frequency. That is the UE-O presented in the paper. The competition among operators can cause some problems. On one hand, the increase of vehicles can result in heavier traffic in the competitive lines and serious waste of vehicle resources. On the other hand, vehicles of less profitable lines are transferred to the competitive lines due to the fixed number of vehicles available for each operator, which results in poor service level of transit on the less profitable lines. Therefore, though operators may have high profits under competitive market (Filippini et al., 2015, Zha et al., 2016), the service levels of less profitable lines are very low and passenger satisfaction is poor. It is obviously not the global optimal solution desired by public authorities.
Competition among operators results in that the actual departure frequency of each line is not the global optimal solution desired by public authorities. Therefore, when public authorities allocate transit lines considering the UE-O, an allocation mechanism avoiding competition as much as possible should be designed to guarantee the service level. In the proposed model, three types of players are considered: the public authority, operators, and passengers. As shown in Fig. 1, the interactions among the three players are depicted as a hierarchical system. In the hierarchical system, the public authority aims to optimize the allocation of transit lines so as to minimize the total transit time of passengers. Operators are assumed to compete with each other by changing line frequency, while passengers have corresponding responses and adaptations on the changing line frequency considering operators’ services. In turn, passenger responses ultimately determine the decision of the public authority and operators. In the proposed model, the optimal frequencies of lines maximizing each operator’s profit on each line allocation scheme is obtained considering the UE-O. The competition on the competitive lines is portrayed as a non-cooperative game by changing the departure frequency of each competitive transit line separately so as to maximize the profit of each operator.
In this paper, the problem is formulated as a set partitioning formulation. Then, a branch-and-price algorithm employing both column generation and branch-and-bound is used to find an optimal solution. Finally, the optimal line allocate plan of Development District of Dalian is obtained by using the proposed model and algorithm.
The paper is organized as follows: Section 2 discusses the main findings of the literature on the operation of transit lines, operation plans optimization, and transit competition models applying game theory. Section 3 provides a formal description of the problem. In Section 4, we specify the model. Section 5 presents the solution approach. An instance is studied in Section 6 and we draw some conclusions in Section 7.
Section snippets
Literature reviews
In this section, we review research in the operation of transit lines, operation plans optimization, and transit competition models applying game theory.
In the literature of transit lines operation, a lot of works are devoted to studying operation policies, e.g., some scholars assessed the benefits and drawbacks of transit deregulation policy in private or public transit systems (Mackie et al., 1995, White, 1995, Preston and Almutairi, 2013, Li et al., 2015, Tang et al., 2012, Zhang et al., 2016
Problem description
There exists a non-cooperative game among operators in sharing-operation of transit lines as shown in Fig. 2. The transit line 1 and line 2 are allocated to operator A, while line 3, line 4 and line 5 are allocated to operator B. Competition exists between operator A and operator B because line 1 and line 3 attract passengers from similar locations. On the competitive lines 1 and 3, operator A tends to deploy more vehicles on line 1 to compete for more passengers, and operator B adopts the same
Notation and assumptions
The model proposed in this paper is intended to be computed at the stage of service planning. It is assumed that each line is only operated by one operator and the number of vehicles available for each operator is fixed. In reality, different vehicle types may be adopted by different public agencies. However, how to efficiently distribute different vehicle types to different transit routes becomes another new problem. To simplify the problem, in this paper, the type of vehicles is assumed to be
Solution method
In the integer model [P1], the number of feasible allocations is so large that it is impractical to solve the model by using mathematical programming software directly. Column generation algorithm is a method to solve such problems having plenty of variables (Barnhart et al., 1998, Gamache et al., 1999, Desrochers et al., 1992). And when solving large-scale integer programming problem, it was proved that column generation was a successful method as a complement of integer programming approach,
Case study
An instance based on the Development District of Dalian are conducted in this section to validate the line allocation model proposed in this paper. The instance is solved by CPLEX 12.5 and in each iteration the subproblem is calculated in Visual studio 2010.
Conclusions
This study focuses on the allocation of transit lines problem in operation-sharing by considering the competition mechanism among operators. While huge literature focuses on the operation of transit lines, few studies incorporated the allocation of transit lines in operation-sharing. Different from the government monopoly, allocating transit lines in operation-sharing reasonably is important for avoiding vicious competition between operators and improving the transit service. This paper
Acknowledgments
This research was supported by the National Natural Science Foundation of China 71571026 and 51578112, Liaoning Excellent Talents in University LR2015008 and the Fundamental Research Funds for the Central Universities (YWF-16-BJ-J-40 and DUT16YQ104)
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This article belongs to the Virtual Special Issue on “Mobility Strats & Effect”.