A train dispatching model based on fuzzy passenger demand forecasting during holidays

Purpose: The train dispatching is a crucial issue in the train operation adjustment when passenger flow outbursts. During holidays, the train dispatching is to meet passenger demand to the greatest extent, and ensure safety, speediness and punctuality of the train operation. In this paper, a fuzzy passenger demand forecasting model is put up, then a train dispatching optimization model is established based on passenger demand so as to evacuate stranded passengers effectively during holidays. Design/methodology/approach: First, the complex features and regularity of passenger flow during holidays are analyzed, and then a fuzzy passenger demand forecasting model is put forward based on the fuzzy set theory and time series theory. Next, the bi-objective of the train dispatching optimization model is to minimize the total operation cost of the train dispatching and unserved passenger volume during holidays. Finally, the validity of this model is illustrated with a case concerned with the Beijing-Shanghai high-speed railway in China. Findings: The case study shows that the fuzzy passenger demand forecasting model can predict outcomes more precisely than ARIMA model. Thus train dispatching optimization plan proves that a small number of trains are able to serve unserved passengers reasonably and effectively. Originality/value: On the basis of the passenger demand predictive values, the train Journal of Industrial Engineering and Management – http://dx.doi.org/10.3926/jiem.699 321 dispatching optimization model is established, which enables train dispatching to meet passenger demand in condition that passenger flow outbursts, so as to maximize passenger demand by offering the optimal operation plan.


Introduction
With the constant improvement of China's high-speed railway network and the enhancement of connectivity within the area of the road network, inter-regional passenger demand increases.
The train can't ensure the safety and punctuality of the train operation, also unable to meet the passenger demand.A large number of passengers are stranded at the station or transfer to other modes of transport to travel.Especially in the traditional holiday, the amount of travel passenger reached a peak value in a short period of time because of the sudden increase in passenger traffic and the uneven distribution.Some trains lack of staff and run without passengers, which causes great waste in transport capacity and increases the total costs of consumption invisible.It's very necessary to develop detailed train dispatching program, distribute and transport passenger planned.
Train dispatching is a multi-variable, multi-constraint, large-scale and multi-objective combination of optimization.In 1971, Amit and Goldfarb applied mathematical programming methods to train scheduling problem.Experts and scholars studied extensively train scheduling problem with mathematical programming theory (Dejax & Crainic, 1987;Beaujon & Turnquist,1991).Scholars did a large number of valuable researches in train deployment method.For example, the concept of all passengers traveler time is proposed (Ghoseiri et al., 2004), which is the index to evaluate passenger satisfaction.He also established dual goal programming model, whose target is to minimize all travelers' time and energy consumption.
Objective optimization model (Higgins et al., 1996;Chen et al.2003), which could minimize the total train delays and operating costs, is established.The mixed integer programming model (Kraay et al., 1991) to minimize train delay and total energy consumption is proposed.The integer programming model (Zhou & Zhong, 2005;Zhang & Jin, 2005) on considering the waiting time in train station and all the train travel time is established.Fuzzy expected value model by fuzzy variables (Yang et al., 2009), which goal is to minimize all passenger travel time and energy consumption, is estimated.It took tabu search algorithm to solve the train scheduling problem.And the objective function is the minimum rate of train delays.(Dong, Wang and Yan, 2005).
Study of train dispatching issues was not able to meet passenger demand, so we must firstly forecast and assess passenger demand.Existing short-term railway passenger demand forecasting contained the relevant model predictions method and time model prediction method.Some researchers established model for passenger demand forecast on considering the transport competition, OD partition, the level of socio-economic development, national income and other factors.The model contained neural network model, Logit model (Peter, Aura & Tommaso, 1996), multiple regression model and the MD model (Butkevicius, Mindaugas, Vladas & Skirmantas, 2004;Fabio, 2006).Time model prediction methods included passenger demand forecasting method based on time series analysis, the gray mode l (Chen, 2008), ARIMA model and BP neural network model (Guo, Qiao, 2008;Wang, 2006).This paper analyzed the holiday passenger flow characteristics based on passenger data from Beijing South Railway Station to Langfang station during 2011 National Day holidays.Then established the passenger change rate of fuzzy logical relationship based on fuzzy theory and time series theory, and proposed fuzzy passenger demand forecasting model.A bi-objective programming model is established and the goal is to minimize the total operation cost of the train dispatching and unserved passenger volume.In this model, passenger demand is known, the constraints are station capacity restrictions, section capacity restrictions, the required number of trains restrictions and train set configuration restrictions.Thus, an interior-penalty method is used to solve this difficult problem.This paper took the Beijing-Shanghai high-speed railway for example.The results showed that the train dispatching optimization model is reasonable and effective, and better able to meet the actual passenger demand.The structure of this paper is as following Figure 1.

Quasi-periodic fluctuation of passenger flow
The running speed of high-speed train is accelerating gradually, which shorten the distance between cities indirectly.The high-speed railway passenger flow is growing year by year during the holiday, daily morning and evening peak traffic is large and flat peak period passenger flow is gentle.The passenger flow quasi-periodic fluctuations are significant.

Regularity of passenger flow
The passenger flow between the two stations in high-speed passenger railway line is statistical by a certain period of time.The history passenger flow in different periods is p(1), p(2),…p(t-1),p(t),p(t+1),…p(n-1), p(n).Take into account the passenger flow change rates between adjacent periods, which are denoted by v(1), v(2),…v(t-1),v(t),v(t+1),…v(n-2), p(n-1).Then analyze the passenger flow change rate, summarize up the regularity of the changes in passenger flow of the adjacent period.
In order to express passenger flow trend in adjacent period clearly and more accurately, passenger flow change rate is normalized.Define standardized passenger flow change rate is  When p(t+1)-p(t)<0, the passenger flow descends from period t to t+1.
In order to reflect the regularity of the passenger flow trend clearly and express varying degrees of passenger flow change respectively, we divide passenger flow change rate into eight intervals applying Zadeh's fuzzy set theory (Zadeh, 1999).
Define the universe of discourse U={u 1 ,u 2 ,u 3 ,u 4 ,u 5 ,u 6 ,u 7 ,u 8 }, and partition it into equal length A 1 denotes that passenger flow decrease is too large, A 2 denotes that passenger flow decrease is larger, A 3 denotes that passenger flow decrease is micro-large, A 4 denotes that passenger flow decrease is less, A 5 denotes that passenger flow increase is less, A 6 denotes that passenger flow increase is micro-large, A 7 denotes that passenger flow increase is larger, A 8 denotes that passenger flow decrease is too large.
Determine the membership function of fuzzy subset A i using assign method, namely: when Define fuzzy subset A i as follows: Establish fuzzy logical relationships based on the fuzzed passenger flow change rates: A j A p " denotes that "if the fuzzed passenger flow rate from period t-1 to t is A j , the fuzzed passenger flow rate from period t to t+1 will be A p ".For example, information of passenger flow is shown in Following the above example, the fuzzy passenger flow change rate for 8:00-9:00 is A 5 , and for 9:00-10:00 is A 3 .Hence, we can establish a fuzzy logical relationship is A 5 A 3 .Similarly, from Table 1, we can get fuzzy logical relationships are A 5 A 3 , A 3 A 5 , A 5 A 4 , A 4 A 8 , etc.
Therefore, the fuzzy logical relationships of the passenger flow change rate are shown in Table 2.
Table 2. Fuzzy logical relationships From Table 2, fuzzy passenger flow change rate is A 5 in previous period, in the following period fuzzy passenger flow change rate are A 3 , A 4 , and A 4 , just as A 5 A 3 , A 5 A 4 and A 5 A 4 .

Fuzzy passenger demand forecasting model
The establishment of fuzzy passenger demand forecasting model is based on fuzzy logical relationships and time series theory, and the steps are as follows:  Step1: Start with period l=n+1 to predict passenger flow. Step3: Calculate the passenger flow change rate in period l=n+1, which is    3 The comparison of prediction model

Train dispatching optimization model
The whole procedure of the model including minimize the total operation cost of the train dispatching and unserved passenger volume.If the operation plan doesn't satisfy the passenger demand, operation plan would be dispatched for passenger flow assignment, in order to attain the passenger demand.

Notation
Input data l: The train type.
T: The planned operating period, i.e., one day.
F l : The variable operating cost for train l running one kilometer.

F ´l:
The variable operating cost for train l empty running one kilometer.
L ij : The distance between stations i and j.
N i,j : The required number of train l between stations i and j. u l : The seat capacity of train l.
U ij : The unserved passengers volume between stations i and j.
C k ij : The carrying capacity between stations i and j on the railway line k.

D:
The fixed overhead cost for one train.
C t : The carrying capacity of station t for the planned operating period T.
C k : The carrying capacity of section k for the planned operating period T.

Decision variables
X i,j,l : The number of train l between stations i and j. y i,j,l : The number of the empty running train l between stations i and j.
x t i,j,l : The number of train l stops at station t between stations i and j. y t i,j,l : The number of the empty running train l stops at station t between stations i and j.
x k i,j,l : The number of train l runs in section k between stations i and j.
y k i,j,l : The number of the empty running train l runs in section k between stations i and j.

Objective function
Train set configuration is to meet the demand of passenger flow outburst.In this case, the minimum total operation cost of the train dispatching and unserved passenger volume are the objective functions to meet the passenger demand.
 Minimize the total operation cost, which includes the fixed overhead cost, variable operating cost for running and variable operating cost for the train empty running.
 Minimize the unserved passengers volume for successfully to meet the passenger demand.

Constraint conditions
 The required number of trains restrictions:

Model of the train dispatching optimization
The train dispatching optimization model is a nonlinear complementarity constraints program.

Case study
In      With the increase in passenger flow during holidays, train dispatching is influenced by more factors and there are more constraints to consider.Further discussion is need for model and more detailed analysis of the railway line is still need in the case.

Figure 1 .
Figure 1.The structure of a train dispatching model based on fuzzy passenger demand forecasting during holidays


Step2: Use Eq.(1) to calculate the passenger flow change rate v(n-1) in period n-1 to n, and fuzzy passenger flow change rate is A i .We can find the next fuzzy passenger flow change rate A j according to fuzzy logical relationships.k i is the number of the passenger flow change rate A j which belongs to passenger flow change rate range i u .
and add predictive value to the data of passenger flow, repeat Step2 to Step3 with regard to l=l+1 until l=M. Step5: Calculate the root mean squared error (RMSE) between the actual values and predictive values, which is flow data are the passenger flow between Beijing and Tianjin in Beijing-Shanghai high-speed railway, and half an hour is an interval between 8:00 and 18:00 from September 21, 2011 to October 30, 2012.720 passenger flow data of first 36 days is historical passenger flow data, 80 passenger flow data of last 4 days is test data.All the computer programs are written in Matlab 7.1.The predictive values and actual values are shown in Figure 3.

Figure 3 .
Figure 3.Comparison of predictive value and real value

Figure 5 .
Figure 5. Line plan of BJ station-JN station in high-speed railway

Table 1 .
DateTime period Passenger flow Change value of passenger flow Change rate Fuzzy set

Table 1 .
Information of passenger flow

Table 3 .
, Train set configuration restrictions: if the round-trip trains are no-load, it can't satisfy the condition of passenger flow outburst between stations i and j.

Table 4 ,
the data is the passenger flow between Beijing and Jinan in Beijing-Shanghai highspeed railway, an hour is an interval between 8:00 and 20:00 from September 27, 2011 to October 10, 2012.All the computer programs are written in Matlab 7.1.The passenger flow predictive values are shown as Figure 4.

Table 4 .
Information of the historical passenger flow data

Table 5 .
Passenger demand forecasting value