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 distancebased 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.
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 (distancebased, sectionbased, and so on).
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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 distancebased 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
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.
The research range is limited to an urban rail line for the independence of operational and fare policy of many urban rail transit lines.
The research time period is a travel time interval of passenger flow (e.g., the morning or evening peak hour).
The operation service of the urban rail line uses a long train route and an allstop schedule. And every train has a uniform number of vehicles.
A transit line
Headway is denoted by
The potential passenger demand and the actual passenger demand between stations
The fare of the two fare structures,
Generalized travel costs of passengers are a weighted combination of fare spending, time costs, and congestion costs. Fare cost is decided by fare structure and its associated fare amounts. Time costs are weighted by wait time at stations and by invehicle time.
When arriving time follows a uniform distribution, the average wait time,
Invehicle time of passengers comprises train operation time and stop time,
Congestion cost is related to the volume of passengers. During period
As shown by the cost equations, except for fare expenses, the passengers’ time and the congestion costs are determined by the train schedule. So passenger flow
The operator’s benefit needs to consider both operating income and operating expenses. Operational income
Operational cost consists of the following three components: train operating cost
The train’s operating cost is directly related to the fleet size
In addition to rervedents of vehicles above. the service cars, the number of reserved and maintenance cars is
Rail line maintenance costs can be expressed as
The total rail stations cost can be expressed as
The train frequency should satisfy the constraint of tracking interval time
Maximum passenger flow of section
Meanwhile, the fare of the whole line should be set for an upper bound limited in operational costs and economic level. The fare of upper bound for all fare structures is
For the problem of urban rail transit route fares, the selection of the optimization objective functions needs to consider the operators’ benefits, the passengers’ benefits, and the social service function. In many literatures, the problem has been studied as a multiobjective program. In the following, the different models will be established according to different optimization objectives in order to compare the effect of these objective functions.
The direct benefit of urban rail transit operators is the train’s operating profit, which can be expressed as the operation revenue (ticket income) minus the operation costs. The objective function of the operator’s direct operation benefit during the study period
Urban rail transit has the dual nature of public welfare and profitability. The overall benefit of an urban rail transit system relates both to the operator and to the passengers. The operator’s benefit can be measured in its fare income. Passenger benefits
Therefore, the objective function of maximum system benefits can be expressed as follows:
Urban rail transit regulators pay attention to the sustainable operation capacity and the traffic service functions of the urban rail transit system. Thus, the operator’s benefit and the volume of passenger turnover
In order to make the two objectives of the operation supervision into a single objective function, the relative weight factor
The cost minimization of a rail transit system can also be used as an optimization objective of fare strategies. The objective function is the sum of two part costs, operators’ costs, and passengers’ generalized travel costs; namely,
Based on the above analysis, models (M1) through (M4) consist of one of the objective functions (
Each of the above models (M1) through (M4) is a nonlinear optimization problem. Therefore, we have designed a SAbased general solution algorithm, for every fare structure in the models. The models (M1) through (M4) can be solved in a uniform solution structure.
In the algorithm, the generation algorithm of the feasible solution and the calculation method for travel OD matrix are each described here.
Generation algorithm of the feasible solution.
Select randomly
Judge whether
If
If all the elements in
Calculation method for OD matrix.
Set
Calculate the passenger flow,
If
In order to realize a satisfied convergence speed, we adopted a very fast generation mechanism of new solution (VFSA; see [
Based on the above analysis, the general GA under every fare strategy is as described below.
The general SA algorithm for all fare strategies.
Initialization. Generate the initial feasible solution
Construction of neighborhood. Generate neighborhood solutions
Metropolis sampling. When
Test of the termination criterion of the inner cycle. If
Cooling schedule. Calculate the temperature
Test of the termination criterion of the outer cycle. When
The first phase project of Metro Line 2 in Changsha, China, of which the total length is 21.36 km and includes 19 stations, is planned to be completed in 2015. In this section, the morning peak hour (7:008:00) of 2016 is chosen as the study period.
The baseline values for parameters are shown in Table
Baseline values for parameters.
Symbol  Value  Unit  Symbol  Value  Unit 


0.15  — 

210  Passengers 

4  — 

1.5  — 

0.49  1/h 

25%  — 

0.1  1/h 

24%  — 

0.05  1/h 

0.8  — 

1.2  1/h 

3800  ¥/km 

325  ¥/hveh 

525  ¥h 

6  Vehicle 

4200  ¥/km 

1/120  h 

0.5  ¥ 

25  min 

2  min 

40  km/h 
In the model of M3, because the objective function of M3 is divided into two parts, the operator part
Optimization results of different models.
Fare structure  Models  

M1  M2  M3  M4  
FF  DBF  FF  DBF  FF  DBF  FF  DBF  
Fare (¥)  5.05  2.00 + 0.28 
8.45  1.98 + 0.27 
3.44  2.00 + 0.05 
8.45  0.35 
Headway (min)  5.27  5.02  11.96  5.20  3.72  3.73  11.90  5.27 
Demand (10^{3} pass.)  57.35  74.04  3.30  75.07  80.78  94.94  3.40  92.75 
Objective function value (10^{3})  59.79  16.83 


299.34  274.62  213.26  296.83 
Fare rate (¥/pass.km)  0.81  0.66  1.68  0.64  0.55  0.38  1.67  0.35 
Maximum section demand (10^{3} pass)  14.34  16.52  0.72  16.85  20.25  23.30  0.75  20.50 
Turnover volume (10^{4} pass.km)  35.97  38.89  1.66  39.91  50.85  58.32  1.72  48.49 
Average riding distance (km)  6.27  5.25  5.03  5.32  6.29  6.14  5.06  5.23 
Maximum passenger density of section (10^{3} pass.km)  2.69  3.47  0.16  3.52  3.78  4.44  0.16  4.34 
Maximum section load rate (%)  94.85  109.23  11.47  111.42  94.56  108.80  9.87  135.57 
Relationship between objective function value and
FF structure
DBF structure
When only the objective function values are taken into consideration, the results of the FF structure are always better than those of the DBF structure. Under Model M1, the objective function value of the FF structure is far better than that of the DBF structure. But for Models M2 to M4, the difference of the objective function value is smaller. The objective function value of Model M2 is negative. This is because the costs of passengers and of the operators are both taken into consideration, especially the effect of the passengers’ generalized travel costs.
The volume of passenger demand and the maximum section load rate between these two fare structures varies enormously in these models. The DBF structure is more attractive to the demand as the fare rate of the FF structure is always higher than that of the DBF structure. Taking the M1 for example, the headway of these two structures is almost equal, but the difference between the objective functions is extraordinarily notable because the difference of the fare rate is significant, 0.15¥/passengerkm.
For the following parameters, the parameter value of the DBF structure is always better than that of the FF structure: the total passenger volume, the turnover volume, the maximum section load rate, the line load, and the maximum sectionpassenger volume. It is clear that the FF structure has an inhibitory effect on passenger flow volume, especially for the shortdistance passengers. Meanwhile, it is indicated that the DBF structure can make full use of the transportation capacity.
The parameter values of average riding distance under Models M2 and M4 are nearly equal. For Models M2 and M4, the average riding distances are shorter than those of the other models, and the average riding distance of the FF structure is shorter than that of the DBF structure. But in Models M1 and M3, the average riding distances of the FF structure are longer than those of Models M2 and M4.
For the fare of the FF structure, Models M1 and M3 are both lower; the fares are ¥5.05 and ¥3.44, respectively. Their average riding distances are similar, while, the lower fare in Model M3 would attract more passengers than would Model M1. In Models M2 and M4, the fares are both ¥8.45, but the values of the two parameters, the maximum section load rate, and passenger demand, are not reasonable. This is because the FF structure cannot correctly reflect the relationship between the riding distance and the passengers’ and operators’ costs and benefits.
For the DBF structure, the fare of M1 almost equals the fare of M2, in which the former is 2.00 + 0.28
Model M1 under the two fare structures and Models M2 and M4 under the DBF structure have similar headways, at about 5.2 minutes.
But the headways of Models M2 and M4 on their own two fare structures are greatly different. Take M2 for example: the headway on the FF structure is 11.96 minutes, and it is 5.2 minutes on the DBF structure. This is because the fare for the DBF structure is fairer than the fare for the FF structure. The DBF structure has an encouraging effect on passenger trips. However, the inhibition effect of the FF structure on passenger volume is more obvious. So the headways in the FF structures of Models M2 and M4 reach a very high level.
For Model M3, the headways on the two fare structures are both about 3.7 minutes. They are lower than the headways of the other models, a difference that reflects the fact that the regulator encourages the operator to provide a high level of service to attract a greater volume of passengers.
For Model M2, the effect of different parameters
Effect of
At the same time, in the case of M1, the spacing distribution of passenger flow is shown in Figure
Distribution of passenger riding distance.
In this paper, for the two typical fare structures (FF and DBF), we present the models of fare decision under different objective functions, respectively, and build a solution algorithm based on GA algorithm. From the comparison and the analysis for the optimization results of multiple models, the following conclusions for Changsha Metro Line 2 can be drawn.
In general, the optimization result of M1 is relatively reasonable in the FF structure, and the fare should be taken as 5.05¥. The fare level in the DBF structure is
The FF structure has an obvious inhibitory effect on shortrange passenger flow, which, generally, is not recommended. The flat fare will be under consideration only when the government would like to supply the enough financial subsidy or economic support so that the low price policy is available.
Considering the fact that the passenger percentage of riding distance from 0 to 6 km accounts for roughly 60% of the total demand, the DBF structure or some of its simplified structures had better be taken to attract more shorthaul passengers.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is supported by the Science and Technology Research Development Program of China Railway Corporation (Major Program, 2013X004A) and the Research Fund for Fok Ying Tong Education Foundation of Hong Kong (132017), the National Natural Science Foundation of China (71471179, 70901076).