A fuzzy optimization model based on improved symmetric tolerance approach is introduced, which allows for rescheduling highspeed railway timetable under unexpected interferences. The model nests different parameters of the soft constraints with uncertainty margin to describe their importance to the optimization purpose and treats the objective in the same manner. Thus a new optimal instrument is expected to achieve a new timetable subject to little slack of constraints. The section between Nanjing and Shanghai, which is the busiest, of BeijingShanghai highspeed rail line in China is used as the simulated measurement. The fuzzy optimization model provides an accurate approximation on train running time and headway time, and hence the results suggest that the number of seriously impacted trains and total delay time can be reduced significantly subject to little cost and risk.
The infrastructures of highspeed railway have been extensively developed in China for the past several years. The network topology structure and operation mode of the railway are changing profoundly. The target is to cover its major economic areas with a highspeed railway network, which consists of four horizontal and four vertical lines [
Train rescheduling in China is mostly manually developed by the operator with the support of computer in the existing line or the high speed railway, which depends on individual's experience to dominate single line or some sections in one line with fixed threehour period. In future, China will establish six comprehensive operation centers of the railway network, each of which will dominate several lines covering one thousand kilometers overall. The scale of rescheduling object is far larger, and the relationship between lines is far more complex than the existing situation, so the inefficient manual rescheduling, which seriously affects the use of the ability of highspeed rail and brings risk to railway safety, cannot support the advanced operation command mode.
The research on the train operation automatic adjustment has been a controversial topic for many years in literature. Variable academic algorithms for optimal rescheduling have been put forward. Some traditional optimization model, such as integer programming (branch and bound method [
Since the railway network has been extensively developed, and the operation mode is becoming more complex in recent years, many heuristic methods, such as DEDSbased simulation method [
Uncertainty optimization theory is to solve the optimization decision problem with all kinds of uncertainties, which involves stochastic optimization, fuzzy optimization, rough set, and so on. Nowadays, some papers studied uncertainty in train scheduling problem. Jia and Zhang [
In this paper, we attempt to achieve optimal timetable rescheduling under the uncertainties, for example, constraints and/or unexpected parameters, by means of proposing the fuzzy optimization model as discussed in the above. In the rest of this paper, typical rescheduling model will be discussed in the section of Timetable Rescheduling Problem. In the following section, we will describe in details of the fuzzy optimization model based on improved tolerance approach to timetable rescheduling, including the fuzzy membership functions of the original objective and soft constraints. A case study on the busiest section of BeijingShanghai high speed line will be illustrated in the section of Case Study. The final section concludes the results of the paper and suggests for further research.
The aim of train rescheduling is to get a new timetable that adjusts the train movements to be consistent with the planned schedule as much as possible under some interference [
Take a rail line with
The decision variables are described as follows:
To minimize the delay cost:
To minimize the number of seriously impacted trains:
We set the final objective function as:
Section running time restrictions:
The real departure time cannot be earlier than the original departure time:
Station dwell time restrictions:
Track restrictions:
Station headway restrictions.
For each station, if two trains use the same track, at least one of
Section headway restrictions.
For each section, at least one of
Auxiliary restrictions:
The headway time for each station is decided by the number of receivingdeparture track, the operation time of turnout, and holding time of the track, which also faces uncertainty problems due to the factors of equipment status and human technological level. So there should be a tolerance for
The headway time for each section is decided by the number and length of the block between two tracking trains and the speeds of the trains, which ranges from 2.4 minutes to 3 minutes according to Shi [
Since the four constraints are changed as above, the model turns to be not representative in the sense of mathematical viewpoints. To construct a reasonable mathematical model under the uncertain environment, the tolerance approach based timetable rescheduling model will be introduced in the next section.
In the paper, we use the fuzzy optimization based improved tolerance approach to solve the uncertainty program, and some necessary backgrounds and notions of the approach are reviewed.
Tolerances are indicated in any technical process, that is, the admissible limit of variation around the object value and the deviations allowed from the specified parameters [
A general model of a fuzzy linear programming problem is presented by the following system [
Here we only discuss the special case (
The
Fuzzy membership functions.
Constraint fuzzy membership function
Objective fuzzy membership function
Fuzzy constraints will inevitably lead to the fuzzy objective based on the ideology of symmetric model. Thus, for the objective function
In order to determine a compromise solution, it is usually assumed that the total satisfaction of a decision maker may be described by
Since we treat the objective in the same manner as the soft constraints, this is also a symmetric model [
Figure
The principle of the fuzzy model.
In the rescheduling model mentioned above, four additional inputs are involved to describe the tolerances of minimum running time, dwell time, and headway time for section and station, respectively. An additional decision variable
Fuzzy membership functions of objective and soft constraints are given by
Since
This model is simulated on the busiest part of Beijing to Shanghai high speed line, between Nanjing (Ning for short) and Shanghai (Hu for short). In the rest of this paper, “HuNing Section” is used to represent this part of the Beijing to Shanghai high speed line. There are seven stations on the HuNing Section, thereby there are 6 sections whose lengths are 65110 m, 61050 m, 56400 m, 26810 m, 31350 m, and 43570 m. Its daily service starts at 6:30 am and ends at 11:30 pm. As currently planned, there are 52 trains (14 high speed trains and 38 medium speed trains) from Beijing to Shanghai line and 8 extra medium speed trains from Riverside line that go from Ning to Hu by the HuNing Section. The trains from Beijing to Shanghai line are called selfline trains, while those from Riverside line are called crossline trains. In reality, selfline trains have higher priority than crossline trains. As for selfline trains, the high speed trains have higher priority than the quasihigh speed trains. Since Beijing to Shanghai high speed line is double track, we only consider the direction from Ning to Hu without loss of generality. The experimental procedure is divided into two stages for the full proof of the validity of the fuzzy model. Firstly, we do research in six aspects with different fuzzy constraints of the same weight, which proves the effectiveness of the tolerancebased fuzzy model in different trains operation conditions. Then a sensitivity analysis of the weighing factors is realized based on the previous operation circumstances. All the models are solved by Ilog Cplex 12.2.
In the simulation, we assume that the train G103 is late for 20 minutes in the section from Nanjing to Zhenjiang, and the trains can run at the normal speed in all the sections; the headway is set from 3 minutes to 2.5 minutes, and the minimum separation time on track possession in each station is set to 1 min.
First we use the original model to solve the problem and get the follow results. When the speed of highspeed train is set to 360 km/h, the speed of medium highspeed train is set to 320 km/h, the headway time is defined as 3 minutes, and the solution value is 368784; there are 3 trains late with the delay time between 10 minutes to 20 minutes and 1 train late with the delay time between 20 minutes to 30 minutes; and the total delay time is 47.5 minutes. When the speed of highspeed train is set to 380 km/h, the speed of medium highspeed train is set to 350 km/h, the headway time is defined as 2.5 minutes, and the solution value is 350449; there are one train late with the delay time between 0 minutes to 10 minutes and 2 trains late with the delay time between 10 minutes to 20 minutes; and the total delay time is 31.85 minutes. Then we use the above results as the inputs of the fuzzy model the paper described before and get the follow results. The fuzzy member is 0.86392, which means the average high speed is 365 km/h, the average medium high speed is 335 km/h, and headway time is 2.9319 minutes; there are also only one train late with the delay time between 0 minutes to 10 minutes and 2 trains late with the delay time between 10 minutes to 20 minutes; and the total delay time is 33.5 minutes. The adjustment strategy is to extend the dwell time for the train K101 and then make the train G105 overtake the train K101 at Changzhou North Station, which reflects the adjustment priority for highgrade train. Figure
Timetable of one train delay.
In the simulation, we assume that the train G305 is late for 30 minutes in the section from Nanjing to Zhenjiang, then the train G103 gets further delay for 20 minutes in the section from Zhenjiang to Changzhou under the condition of the existing delay, and other conditions are identical to Section
First we use the original model to solve the problem and get the follow results. When we take the strict constraints, the solution value is 860720; there are 2 trains late with the delay time between 0 minutes to 10 minutes, 3 trains late with the delay time between 10 minutes to 20 minutes, and 3 trains late with the delay time between 20 minutes to 30 minutes; the total delay time is 122.43 minutes. When we relax the constraints, the solution value is 848871; there are 2 trains late with the delay time between 0 minutes to 10 minutes, 2 trains late with the delay time between 10 minutes to 20 minutes, and also 3 trains late with the delay time between 20 minutes to 30 minutes; and the total delay time is 109.18 minutes. Then we solve the fuzzy model and get the follow results. The fuzzy member is 0.89425, which means the average high speed is 362 km/h, the average medium high speed is 332 km/h, and headway time is 2.9512 minutes; there are also 7 delay trains, 2 trains of which late with the delay time between 0 minutes to 10 minutes, 2 trains late with the delay time between 10 minutes to 20 minutes, and 3 trains late with the delay time between 20 minutes to 30 minutes; and the total delay time is 114.29 minutes. The adjustment strategy is to make the high level train G105 subsequently overtakes the train K115 L15 and K101 at Zhenjiang West Station, and train L15 overtakes the train G103 at the Wuxi East Station, which effectively avoid the high level train G105 being late and reduce the delay time of the train L15. Figure
Timetable of two trains delay.
When some natural hazards happened, like heavy storm and strong wind, railway will be greatly affected in a large area. In the simulation, we assume there is a speed restriction in all sections and the average limited speed ranges from 170 km/h to 150 km/h; other conditions are identical to Section
First we use the original model to solve the problem and get the follow results. When we set the speed as 150 km/h and headway time as 3 minutes, the solution value is 7969213; there are 2 trains late for the time between 30 and 40 minutes, 18 trains late for the time between 40 and 50 minutes, 34 trains late for the time between 50 and 60 minutes, and 6 trains late over one hour; and the total delay time is 3052.95 minutes. When the speed is set to 170 km/h, and headway time is set to 2.5 minutes, the solution value is 7473849; there are 5 trains late for the time between 20 and 30 minutes, 38 trains late for the time between 30 and 40 minutes, 17 trains late for the time between 40 and 50 minutes, and no train late over one hour; and the total delay time is 2248.26 minutes. Although the latter result is very exciting, it is a significant risk to take the speed of 170 km/h, as the highest speed must bring the high operation cost and may arouse some new delays. Then we use the above results as the inputs of the fuzzy model the paper described before and get the follow results. The fuzzy member is 0.689117, which means the average speed is 156 km/h, and headway time is 2.8445 minutes; there are 19 trains late for the time between 30 and 40 minutes, 25 trains late for the time between 40 and 50 minutes, 16 trains late for the time between 50 and 60 minutes, and no train late over one hour; and the total delay time is 2655.16 minutes. We can see that the fuzzy optimization result improved greatly with little slack of constraints that means little risk and operation cost. Figure
Timetable of speed restriction in all sections.
Sometimes equipment failure, like train signal failure, happens in some but not all the sections. In the simulation, we assume there is a speed restriction in first section, and the limited speed ranges from 60 km/h to 50 km/h; other conditions are identical to Section
First we use the original model to solve the problem and get the follow result. When we take the strict constraints, the solution value is 9294339; there are 19 trains late for the time between 40 and 50 minutes, 30 trains late for the time between 50 and 60 minutes, and 11 trains late over one hour; and the total delay time is 3269.32 minutes. When the speed is 60 km/h, and headway time is 2.5 minutes, the solution value is 8577661; there are 20 trains late for the time between 30 and 40 minutes, 37 trains late for the time between 40 and 50 minutes, 3 trains late for the time between 50 and 60 minutes, and no train late over one hour; and the total delay time is 2486.32 minutes. Then we get the follow result using the fuzzy model the paper described before. The fuzzy member is 0.70606, which means the average speed is 53 km/h, and headway time is 2.85303 minutes; there are 5 trains late for the time between 30 and 40 minutes, 39 trains late for the time between 40 and 50 minutes, 16 trains late for the time between 50 and 60, minutes and no train late over one hour; the total delay time is 2753.17 minutes. The adjustment strategy is to make the train G117 overtakes the train K105 at Wuxi East Station, and then let the train L7 overtake the train G143 at the Wuxi East Station. Figure
Timetable of speed restriction in first section.
In the simulation, we assume that there is a speed restriction in first section, and the limited speed ranges from 60 km/h to 50 km/h; the train G103 gets delay in the section from Zhenjiang to Changzhou. Other conditions are identical to Section
First we use the original model to solve the problem and get the following results. When the speed in the first section is 50 km/h; the trains can run at the normal speed; the headway time is set to 3 minutes, the solution value is 9367637; there are 19 trains late with the delay time between 40 minutes to 50 minutes, 24 trains late with the delay time between 50 minutes to 60 minutes, and 17 trains late for more than one hour; and the total delay time is 3305.19 minutes. When the speed in the first section is 60 km/h; the headway time is set to 2.5 minutes; then the solution value is 8671072; 3 trains late with the delay time between 20 minutes to 30 minutes, 34 trains late with the delay time between 30 minutes to 40 minutes, 17 trains late with the delay time between 40 minutes to 50 minutes, 1 train late with the delay time between 50 minutes to 60 minutes, and 5 train late with the delay time more than one hour; and the total delay time is 2557.32 minutes. Then we use the above results as the inputs of the fuzzy model the paper described before and get the follow results. The fuzzy member is 0.740856, which means the average speed in the first section is 52 km/h, and headway time is 2.86 minutes; there are 17 trains late with the delay time between 30 minutes to 40 minutes, 23 trains late with the delay time between 40 minutes to 50 minutes, 15 trains late with the delay time between 50 minutes to 60 minutes, and 5 trains late with the delay time more than one hour; and the total delay time is 2823.34 minutes. The adjustment strategy is to make the trains G105, K115, and L15 overtake the train K115 at Changzhou North Station, and then let the train G105 overtakes the train K101 at the Wuxi East Station. Figure
Timetable of speed restriction in one section and one train delay.
In the simulation, we assume that the train G305 is late for 40 minutes in the section between Nanjing to Zhenjiang and then get further delay for 15 minutes in the section between Zhenjiang to Changzhou; the train G103 gets a delay for 30 minutes in the section from Zhenjiang to Changzhou; also there is a speed restriction in all sections, and the average limited speed ranges from 170 km/h to 150 km/h for all the trains; other conditions are identical to Section
First we use the original model to solve the problem and get the follow results. When we set the speed as 150 km/h and headway time as 3 minutes, the solution value is 8042152; there are 1 train late with the delay time between 30 minutes to 40 minutes, 17 trains late with the delay time between 40 minutes to 50 minutes, 27 trains late with the delay time between 50 minutes to 60 minutes, and 15 trains late with the delay time more than one hour; and the total delay time is 3197.86 minutes. When the speed is 170 km/h, and headway time is 2.5 minutes, the solution value is 7588658; there are 5 trains late with the delay time between 20 minutes to 30 minutes, 35 trains late with the delay time between 30 minutes to 40 minutes, 14 trains late with the delay time between 40 minutes to 50 minutes, 3 trains late with the delay time between 50 minutes to 60 minutes, and 3 trains late with the delay time more than one hour; and the total delay time is 2397.15 minutes. Then we use the above results as the inputs of the fuzzy model the paper described before and get the follow results. The fuzzy member is 0.792877, which means the average speed is 153 km/h, and headway time is 2.89 minutes; there are 17 trains late with the delay time between 30 minutes to 40 minutes, 23 trains late with the delay time between 40 minutes to 50 minutes, 15 trains late with the delay time between 50 minutes to 60 minutes, and 5 trains late with the delay time more than one hour; and the total delay time is 2704.43 minutes. The adjustment strategy is to make the train L1 overtakes the train G101 at the Wuxi East Station, the train L15 and the train K101 overtake K115, and then extend the dwell time of K115 at the Wuxi East Station for the stopover by the train G101, so the importance of the high level trains is also shown from the steps above. Figure
Timetable of speed restriction in all sections and two trains delay.
These simulations are realized on the computer with Intel Core 2 Duo CPU E7500 and 2 sG Memory. All the optimization results for the 6 cases are given in Table
Rescheduling results of different cases.
Case  R  D1  D2  D3  D4  D5  D6  D7  D8  D9  D10  D11 

R1  0  3  1  0  0  0  0  368784  47.5  —  5.38  
Case 1  R2  1  2  0  0  0  0  0  350449  31.5  —  3.39 
R3  1  2  0  0  0  0  0  0.86392  33.5  1  178.17  
 
R1  2  3  3  0  0  0  0  860720  122.43  —  6.22  
Case 2  R2  2  2  3  0  0  0  0  848871  109.18  —  3.36 
R3  2  2  3  0  0  0  0  0.89425  114.29  2  142.58  
 
R1  0  0  0  2  18  34  6  7969213  3052.95  —  3.38  
Case 3  R2  0  0  5  38  17  0  0  7473849  2248.26  —  3.32 
R3  0  0  0  19  25  16  0  0.689117  2655.16  —  23.13  
 
R1  0  0  0  0  19  30  11  9294339  3269.32  —  3.36  
Case 4  R2  0  0  0  20  37  3  0  8577661  2486.32  —  3.39 
R3  0  0  0  5  39  16  0  0.70606  2753.17  2  76.75  
 
R1  0  0  0  0  19  24  17  9367637  3305.19  —  8.05  
Case 5  R2  0  0  3  34  17  1  5  8671072  2557.32  —  7.97 
R3  0  0  0  17  23  15  5  0.740856  2823.34  3  280.23  
 
R1  0  0  0  1  17  27  15  8086249  3197.86  —  7.74  
Case 6  R2  0  0  5  35  14  3  3  7588658  2397.15  —  7.75 
R3  0  0  0  17  23  15  5  0.792877  2704.43  4  260.56 
There are four kinds of fuzzy constraints as described in Sections
Table
Comparison of the optimal objectives.
Case 



0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  
Case 1  0.860180  0.860920  0.861390  0.862250  0.863920  0.864470  0.865731  0.865020  0.864170 
Case 2  0.889860  0.890440  0.891050  0.892280  0.894240  0.896030  0.895490  0.892661  0.890040 
Case 3  0.671015  0.674356  0.678921  0.684357  0.689117  0.693258  0.698091  0.702655  0.705186 
Case 4  0.697540  0.698920  0.700130  0.702480  0.706060  0.709138  0.712240  0.714521  0.715390 
Case 5  0.730050  0.732160  0.734875  0.737280  0.740856  0.744030  0.747980  0.749721  0.751060 
Case 6  0.779810  0.781530  0.784010  0.787903  0.792877  0.797460  0.801130  0.804655  0.806515 
Optimal objectives with different weights.
This paper presents a fuzzy optimization model based on improved tolerance approach for train rescheduling in case of train delay and speed restriction, which deals with the train running time at sections, the headway time at sections, and station as the fuzzy parameters. The simulations on Beijing to Shanghai high speed line reveal that the fuzzy optimization result improved greatly with little slack of constraints. This means we can get a new timetable with less total delay time as well as the number of seriously impacted trains in safe and lower average speed, little dwell time, and enough headway time. In addition, the sensitivity analysis of the weighing factors shows that the same constraints contribute differently to the optimal objective in different emergencies, thus the dispatchers should take different trains adjustment strategies to eliminate interference as much as possible.
There remain many interesting areas to explore around the uncertainty in timetable rescheduling problem. Firstly, the membership functions of fuzzy parameters used in the paper may be more complex form in practice than the linear function in the paper, so we can take some genetic functions, like gauss membership function, to model the fuzzy programming. The more accurate the membership function is, the better result the fuzzy optimization model gets. Secondly, the tolerance can also be described by fuzzy setbased schemes. Finally, fuzzy operators in the model can be improved to adapt well to the information processing mechanism of despatchers in dealing with rescheduling problems. Our ultimate goal is to develop a realtime rescheduling system to significantly improve operation management and scheduling efficiency in the future.
This work has been supported by National Natural Science Foundation of China (61074151); National Key Technology Research and Development Program (2009BAG12A10); Research Found of State Key Laboratory of Rail Traffic Control and Safety (RCS2009ZT002, RCS2010ZZ002, and RCS2011ZZ004).