Fare Optimality Analysis of Urban Rail Transit under Various Objective Functions

Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2014, Article ID 910736, 8 pages http://dx.doi.org/10.1155/2014/910736 Research Article Fare Optimality Analysis of Urban Rail Transit under Various Objective Functions Lianbo Deng, Zhao Zhang, Kangni Liu, Wenliang Zhou, and Junfeng Ma School of Traffic and Transportation Engineering, Central South University, Changsha 410075, China Correspondence should be addressed to Wenliang Zhou; zwl Received 18 July 2014; Accepted 15 August 2014; Published 28 August 2014 Academic Editor: Wuhong Wang Copyright © 2014 Lianbo Deng 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. Urban rail transit fare strategies include fare structures and fare levels. We propose a rail transit line fare decision based on an operating plan that falls under elastic demand. Combined with the train operation plan, considering flat fare and distance-based fare, and based on the benefit analysis of both passenger flow and operating enterprises, we construct the objective functions and build an optimization model in terms of the operators’ interests, the system’s efficiency, system regulation goals, and the system costs. The solving algorithm based on the simulated annealing algorithm is established. Using as an example the Changsha Metro Line 2, we analyzed the optimized results of different models under the two fare structures system. Finally the recommendations of fare strategies are given. 1. Introduction In the urban rail system, fare strategies include fare structures and fare levels. Fare structures are the relationship between the fare amount and the trains’ travel distance, which includes the flat fare and the graduated fare (distance-based, sectionbased, and so on). Savage [1] found from conducting a time-series analysis of the bus operations of the Chicago Transit Authority from 1953 to 2005 that Chicago could improve social welfare by reducing service frequencies and then by using the saved money to lower fares for a given budget constraint. Using the city of Haifa, Israel, as a case study, Sharaby and Shiftan [2] focused on evaluating the impact of fare integration on travel behavior and on transit ridership, which showed a significant increase in passenger demand and ticket sales when a simple fare system with free transfers, reducing fares for many passengers was adopted. They found that fare reduction was a significant factor in attracting transit users. Litman [3] studied transit price elasticity and cross-elasticity wherein he evaluated public transit benefits and costs (see also, [4]). Some studies [5–7] showed that fare systems, service levels, living standards, and travel demands would affect public transit ridership. Winston and Maheshri [8] found, through estimating the contribution of each US urban rail operation to social welfare based on the demand for and cost of its service, that with the exception of BART in the San Francisco Bay area, every system actually reduces welfare and was unable to become socially desirable even with optimal pricing or with a physical restructuring of its network. Li et al. [9] developed a profit maximization model to optimize rail line length, number and the locations of stations, the headway, and the fares. Chien and Tsai [10] studied optimization of fare structure and service frequency for maximum profitability of a rail line, while peak period and off-peak period were considered, and they conducted sensitivity analyses of fares and headways. Borndörfer et al. [11] studied models for fare planning in public transport, and the study included some objectives such as the maximization of demand, revenue, profit, or social profit, and they proposed a nonlinear optimization approach based on a detailed discrete choice model of user behavior. Then they used the resulting models to compute and to compare optimized fare systems for the city of Potsdam, Germany. Lam and Zhou [12] presented a bilevel model to optimize the fare structure for transit networks with elastic demand under the assumption of fixed transit service frequency, where the upper-level problem seeks to maximize the operator’s revenue, whereas the lower-level problem is a stochastic user equilibrium transit assignment model with capacity constraints. Zhou et 2 Discrete Dynamics in Nature and Society al. [13] also built a bilevel transit fare equilibrium model for a deregulated transit system, where the upper-level problem is to maximize the profit of each transit operator within an oligopolistic market for there exists a generalized Nash game between transit operators, and the lower-level problem is stochastic user equilibrium assignment model with elastic OD demand. Obviously, the fare decision of urban rail transit systems is a multiple objective problem. Studies from various objective functions have shown the differences of fare strategies. In this paper, we present the models of fare strategy under some objective functions, including the two typical fare structures, flat fare (FF) and distance-based fare (DBF). Then we compare the optimal solutions of these models. The remainder of this paper is organized as follows. In the next section, we analyze the fare decision problem, which includes the generalized travel costs of passengers and the operator’s benefits. In Section 3 we discuss the constraints and objective functions and present the optimization models with various objective functions. In Section 4, we develop a solution algorithm based on a simulated annealing (SA) algorithm. In Section 5, the case of Changsha Metro Line 2 is used to illustrate the application of the proposed models and of the solution algorithm. In particular, we analyze and discuss the solutions of each fare structure under objective functions. Finally, the conclusions and a recommended fare policy are given in Section 6. 2. Problem Statements Urban passenger flow typically exhibits an obvious characteristic of elastic demand that is affected by generalized travel costs determined by the train operation organization. So the fare of urban rail transit must be optimized comprehensively by combining with the train schedule. For simplification, the following assumptions are made in this paper. (A1) The research range is limited to an urban rail line for the independence of operational and fare policy of many urban rail transit lines. (A2) The research time period is a travel time interval of passenger flow (e.g., the morning or evening peak hour). (A3) The operation service of the urban rail line uses a long train route and an all-stop schedule. And every train has a uniform number of vehicles. A transit line 𝑙(𝑁) = (𝑆, 𝐸) is represented by an ordered sequence of stations 𝑆 = {1, 2, . . . , 𝐿 𝑆 }, and 1, 2, . . . , 𝐿 𝑆 is arranged by the down direction. The (...truncated)