CN107284480B - A kind of automatic preparation method of route map of train based on the multiplexing of vehicle bottom - Google Patents

A kind of automatic preparation method of route map of train based on the multiplexing of vehicle bottom Download PDF

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CN107284480B
CN107284480B CN201710428127.7A CN201710428127A CN107284480B CN 107284480 B CN107284480 B CN 107284480B CN 201710428127 A CN201710428127 A CN 201710428127A CN 107284480 B CN107284480 B CN 107284480B
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CN107284480A (en
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王义惠
宁滨
唐涛
宿帅
廖志斌
张瞄
罗依梦
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Beijing Jiaotong University
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Abstract

The embodiment of the invention provides a kind of automatic preparing methods of route map of train based on the multiplexing of vehicle bottom, are related to Train Operation Control Technology field, comprising the following steps: underlying parameter needed for configuration establishment route map of train；According to the basic parameter configuration train running scheme；Route map of train Optimized model is established according to the underlying parameter and train running scheme；The route map of train Optimized model is solved using mixed integer linear programming algorithm, obtains the route map of train based on the number of allocated passenger trains amount to put into effect.The present invention improves the establishment efficiency of operation figure, automatically the route map of train worked out can meet the constraint conditions such as reciprocal time, personal distance, train sequence, rolling stock, guarantee the feasibility of operation figure, it cuts operating costs, passenger waiting time is reduced, the needs that urban rail system fine-grained management Train operation figure frequently adjusts can be met.

Description

Train running diagram automatic compilation method based on vehicle bottom reuse
Technical Field
The invention relates to the technical field of train operation control, in particular to an automatic compilation method of a train operation diagram based on vehicle bottom multiplexing.
Background
With the continuous development of urban rail transit systems, the road network structure gradually becomes more and more complex. By 7 months in 2015, the total planned mileage of urban rail transit approved by national institutes exceeds 7300 kilometers, and by 2020, the total mileage of urban rail transit can reach 7395 kilometers.
Under the condition that the development scale of urban rail transit is gradually huge, the compilation of a train operation diagram as a key technology of urban rail transit operation management plays an important role, and the implementation quality directly influences each link of rail transit operation.
Meanwhile, the application of the train bottom is closely connected with the operation of urban rail transit, in a running chart, the number of the train bottoms which are put into operation is directly influenced by the size of the departure intervals of the trains, and whether the trains turn back at a terminal station or not also directly influences the arrival and departure time of each station.
On the other hand, the conventional train diagram generation method does not take the train bottom use into consideration. In view of this, in order to improve the operation management level of urban rail transit, an optimization method considering train bottom application is urgently needed to achieve automatic compilation of a train operation diagram.
Disclosure of Invention
The embodiment of the invention provides the automatic compilation method of the train running chart based on the vehicle bottom reuse, which can improve the compilation efficiency of the train running chart, meet the requirement of the urban rail system on the fine management, improve the feasibility of the train running chart, reduce the train operation cost and reduce the waiting time of passengers.
In order to achieve the purpose, the invention adopts the following technical scheme.
A train operation diagram automatic compilation method based on vehicle bottom reuse comprises the following steps:
configuring basic parameters required for compiling a train operation diagram;
configuring a train running scheme according to the basic parameters;
establishing a train operation diagram optimization model according to the basic parameters and the train operation scheme, wherein the train operation diagram optimization model comprises the following steps: the train dispatching interval calculation model, the train dispatching moment calculation model, the train sequence constraint model, the train bottom application constraint model, the objective function model and the train circulation plan calculation model;
and solving the train operation diagram optimization model by adopting a mixed integer linear programming algorithm to obtain a train operation diagram based on vehicle bottom multiplexing.
Further, the basic parameters include:
basic parameters needed for compiling a train running diagram are configured according to an actual train line and an operation condition, wherein the basic parameters comprise line information, station running time, in-station parking time, train operation starting and stopping time, first and last bus time points, a minimum departure interval, a maximum departure interval, a minimum turn-back time and a maximum turn-back time;
there are J stations on city rail road, denoted as SstaJ, and the vehicle section is connected to station 1, and defines an operation direction from station 1 to station J as an up direction and an operation direction from station J to station 1 as a down direction.
Further, the configuring the train operation scheme includes:
starting and stopping time [ t ] of train operationstart，tend]Divided into K time segments, i.e. T1＝[t0，t1]，T2＝[t1，t2]，…，Tk＝[tK-1，tK]Wherein t is0＝tstart，tK＝tend；
Each time period is required to finish IkNumber of individual vehicles with total number of vehiclesThe set of the train numbers is Sservice＝{1，2，...，Itotal}。
Further, the train departure interval calculation model includes:
every two adjacent trains have departure intervals in each time period, and the departure interval of the Kth time period is as follows:and isWherein h ismin、hmaxRespectively representing a minimum departure interval and a maximum departure interval.
Further, the train arrival time calculation model includes:
the arrival and departure time of the ascending train at each platform has the following calculation formula:
wherein,represents the ascending departure time of the train number i at the station j,representing the time of arrival of the train number i at the station j,represents the up-going stop time of the train number i at the station j,representing the uplink running time of the train number i from the station j to the next station j + 1;
the arrival and departure time of the descending train at each platform has the following calculation formula:
wherein,indicating the time of departure of the train number i at the station j,indicating the time of arrival of the train at station j at train number i,indicating the down stop time of train number i at station j,representing the downlink running time of the train number i from the station j to the next station j-1;
the departure time of the first and last trains is obtained according to the time points of the first and last trains, and the calculation formula is as follows,
departure time of the first train at the starting station:
departure time of the last train at the starting station:
wherein, the arrival and departure time of two adjacent trains meets the conditionSo as to ensure that only one train is on the return line at the same moment;
if the train turns back after the station J is the station, the turn-back time of the train at the station JThe following conditions are satisfied:
wherein,andrespectively representing the minimum turn-back time and the maximum turn-back time of the train at a station J;
further, the train sequence constraint model includes:
based on the basic principle of train sequence constraint, the two-train departure time of the train at the station 1 and the train at the station J meets the following constraint:
further, the train circulation planning calculation model comprises:
the train circulation plan calculation method comprises the following steps:
αiindicates whether the train number i is issued by a vehicle segment:
βiindicates whether the train number i returns to the vehicle section:
γi，i′whether the train number i and the train number i' use the same train bottom is represented:
further, the train circulation plan calculation meets the following constraint conditions:
whether the train number i is sent out by the train section depends on whether the train number i' and the previous train number use the same train bottom, namely the relationship is satisfied:
whether the train number i returns to the train section depends on whether the train number i' exists behind and whether the train number i uses the same train bottom, namely the relationship is satisfied:
when the train number i and the train number i 'use the same train bottom, the train number i does not return to the train section, and the train number i' is not sent out by the train section, namely alphai′、βiAnd gammai，i′The following relationship is satisfied:
αi′+βi≤2+M(1-γi，i′)
αi′+βi≥2-M(1-γi，i′)；
if the train turns back after the station 1 is the station, the arrival time of two trains i, i '(i is the front train and i' is the rear train) before and after the station 1 should satisfy the following relationship:
wherein,andrespectively representing the minimum turn-back time and the maximum turn-back time of the train at the station 1; the arrival time of two adjacent trains at the station 1 meets the following turn-back condition:
further, the objective function model includes:
minimum sizeNormalizing an objective functionWherein, w1、w2And w3Is a weight, fobj，1，nomNominal value of departure interval deviation, f, provided for mode of serviceobj，2，nomFor using the number of trains at the same bottom, fobj，3，nomA penalty value for interval changes;
in the objective function fobj，1Is a portion that minimizes departure interval deviation from service mode provision, which is expressed as:
in the objective function fobj，2The number of cars using the same car bottom in the operation stage is maximized, which is expressed as:
in the objective function fobj，3Is a part that limits the variation of the interval of the adjacent train, which is represented by the absolute value of the difference between the current interval and the average value of the interval of the adjacent train, that is:
further, the method for solving the train operation diagram optimization model by adopting the mixed integer linear programming algorithm to obtain the train operation diagram based on the number of train bottoms put into operation comprises the following steps:
step 1, linearizing the real number multiplied by an integer term: in the retrace condition constraint satisfied by the arrival time of the two adjacent trains at the station 1Partially linearized byWherein,
zi，i′≤tKγi，i′，zi，i′≥t0γi，i′
step 2, pair fobj，1The absolute value part of (a) is linearized: the concrete implementation method is
Wherein,
step 3, pair fobj，3The absolute value part of (a) is linearized: the concrete implementation method is
Wherein,
and 4, outputting an optimal solution: inputting configuration parameters, solving the train working diagram optimization model by adopting a mixed integer linear programming algorithm, and drawing a train working diagram in an auxiliary mode.
The invention has the beneficial effects that: the embodiment of the invention improves the compilation efficiency of the operation diagram, the automatically compiled train operation diagram can meet the constraint conditions of turn-back time, safety interval, train sequence, train bottom application and the like, the feasibility of the operation diagram is ensured, the operation cost is reduced, the waiting time of passengers is reduced, and the requirement of frequent adjustment of the train operation diagram under the fine management of an urban rail system can be met.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings needed to be used in the description of the embodiments will be briefly introduced below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings based on these drawings without inventive labor.
Fig. 1 is a processing flow chart of a train running chart automatic compilation method based on vehicle bottom reuse according to an embodiment of the present invention;
FIG. 2 is a diagram of a train operation constructed in accordance with a method of an embodiment of the present invention;
FIG. 3 is a bottom use view of a vehicle constructed in accordance with the method of the present invention;
FIG. 4 is a diagram of an operating interval between an ascending train and a descending train constructed in accordance with the method of the present invention;
FIG. 5 is a diagram of a train operation constructed in accordance with a method of an embodiment of the present invention;
FIG. 6 is a bottom use view of a vehicle constructed in accordance with the method of the present invention;
fig. 7 is a diagram of an operating interval between an ascending train and a descending train constructed according to the method of the embodiment of the invention.
Detailed Description
Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like reference numerals refer to the same or similar elements or elements having the same or similar function throughout. The embodiments described below with reference to the accompanying drawings are illustrative only for the purpose of explaining the present invention, and are not to be construed as limiting the present invention.
For the convenience of understanding the embodiments of the present invention, the following description will be further explained by taking several specific embodiments as examples in conjunction with the drawings, and the embodiments are not to be construed as limiting the embodiments of the present invention.
Example one
The embodiment provides a processing flow chart of an automatic train diagram compiling method based on vehicle bottom multiplexing, which is shown in fig. 1 and mainly comprises the following steps:
step S110: configuring actual line basic parameters required by drawing, comprising: the system comprises line information, station operating time, station stopping time, starting and stopping time of train operation, first and last bus time points, minimum departure intervals and maximum departure intervals allowed by the system, and minimum turn-back time and maximum turn-back time allowed by the system. Such as: there are J stations on city rail road, denoted as SstaJ, and the vehicle section is connected to station 1, and defines an operation direction from station 1 to station J as an up direction and an operation direction from station J to station 1 as a down direction.
Step S120: and configuring a train running scheme according to the basic parameters, namely configuring the number of finished trains in the operation time period.
Starting and stopping time [ t ] of train operationstart，tend]Divided into K time segments, i.e. T1＝[t0，t1]，T2＝[t1，t2]，…，Tk＝[tK-1，tK]Wherein t is0＝tstart，tK＝tend(ii) a Each time period is required to finish IkNumber of individual vehicles with total number of vehiclesThe set of the train numbers is Sservice＝{1，2，...，Itotal}。
Step S130: establishing a train operation diagram optimization model according to the basic parameters and the provided train number, wherein the optimization model comprises the following five sub-models of C1, C2, C3, C3 and C5,
c1 trainCalculating departure intervals: dividing the urban rail line operation period into K time periods according to the step 2, wherein K belongs to {1, 2., K }, and then the kth time period [ t [ ]k-1，tk]The departure interval of (a) is calculated by the following formula:
and satisfy
C2, calculating the arrival and departure time of the train: if the train is in the uplink direction on the urban rail line, the arrival time of the train is calculated as follows:
wherein,represents the ascending departure time of the train number i at the station j,representing the time of arrival of the train number i at the station j,represents the up-going stop time of the train number i at the station j,representing the uplink running time of the train number i from the station j to the next station j + 1;
if the train is in the descending direction of the urban rail line, the arrival time of the train is calculated as follows:
wherein, thereinIndicating the time of departure of the train number i at the station j,indicating the time of arrival of the train at station j at train number i,indicating the down stop time of train number i at station j,representing the downlink running time of the train number i from the station j to the next station j-1;
and the operation start-stop time t of the urban rail line is knownstart，tend]Then, the departure time of the first train and the last train at the starting station is calculated as follows:
the arrival time of two connected trains meets the following conditions:
if the train turns back after the station J is the station, the turn-back time of the train at the station JShould satisfy the following requirementsThe following conditions:
wherein,the requirements of minimum turn-back time and maximum turn-back time need to be met;
and C3, train sequence constraint, wherein the sequence constraint principle for two continuous trains is that the overtaking is not allowed, namely after the train is dispatched from the station 1, the train which is dispatched from the back can not overtake the train which is dispatched from the front. Then at station 1 and station J, the departure times of both cars should satisfy the following constraints:
c4, vehicle bottom application constraint, three binary variables alpha are introducedi，βi，γi，i′To show different states of the train in the train section in relation to the use of the train at the bottom, wherein,
αiindicates whether the train number i is issued by a vehicle segment:
βiindicates whether the train number i returns to the vehicle section:
γi，i′whether the train number i and the train number i' use the same train bottom is represented:
whether the train number i 'is sent by the train section depends on whether the train number i' and a certain train in front use the same train bottom, namely the same train in the physical sense, and the following relations are required to be met:
whether the train number i returns to the train section depends on whether the train number i' exists behind and whether the train number i uses the same train bottom, namely the relationship is satisfied:
when the train number i and the train number i 'use the same train bottom, the train number i does not return to the train section, and the train number i' is not sent out by the train section, namely alphai′、βiAnd gammai，i′The following relationship is satisfied:
αi′+βi≤2+M(1-γi，i′)
αi′+βi≥2-M(1-γi，i′)，
if the train turns back after the station 1 is the station, the arrival time of two trains i, i '(i is the front train and i' is the rear train) before and after the station 1 should satisfy the following relationship:
wherein,andrespectively representing the minimum and maximum turnaround times of the train at station 1.
And the arrival time of two connected trains at the station 1 should satisfy the following turn-back condition:
c5, objective function model: considering the influence of train departure intervals and train circulation plans on the operation diagram, the objective function model minimizes the following function:
wherein, w1、w2And w3Is a weight, fobj，1，nomNominal value of departure interval deviation, f, provided for mode of serviceobj，2，nomIs composed of the number of two trains using the same vehicle bottom, fobj，3，nomA penalty for interval changes.
In the objective function fobj，1Is a portion that minimizes departure interval deviation from service mode provision, which is expressed as:
in the objective function fobj，2The number of cars put into operation in the operation stage is minimized, in other words, the number of cars using the same car bottom in the operation stage is maximized, which is expressed as:
in the objective function fobj，3Is a part that limits the variation of the interval of the adjacent train, which is represented by the absolute value of the difference between the current interval and the average value of the interval of the adjacent train, that is:
in the above optimization models of C1, C2, C3, C4 and C5, C1 to C4 are constraints of the optimization problem, and C5 is an objective function of the optimization problem.
Step S140: and solving the optimization model by adopting mixed integer linear programming to obtain a train operation diagram related to train bottom application.
The above algorithm can adopt MATLAB and CPLEX to realize solution.
Example two
In the implementation process of the method of the embodiment of the invention, the maximum and minimum operation intervals are set as 660s and 120s, and the weight values are respectively 4-103，20，5·104The number of the vehicles to be completed in each time period and the ideal operation interval are shown in the table I and the table II:
watch 1
Watch two
According to the steps, an optimization model for automatically compiling the train operation diagram is established, the problem is solved by adopting a mixed integer linear programming algorithm, the obtained train operation diagram is shown in figures 2 and 5, the train bottom use diagram is shown in figures 3 and 6, and the operation interval of the up-train and the down-train is shown in figures 4 and 7.
In fig. 2, the abscissa represents the operation time of the urban rail line part from 5 am to 10 am, and the ordinate represents the station, and the train number marked by the same line type in the figure represents that the same train bottom is used. In fig. 3, the abscissa is time and the ordinate is the number of the used train bottoms, wherein the squares of the same horizontal line indicate the number of the trains using the same train bottom. In fig. 4, the abscissa represents the number of train operations, the ordinate represents the running interval (in seconds) of the train, and the dots of the dot marks and the square marks in the figure represent the running intervals of the current train and the previous train in the up-and down-going directions, respectively. The horizontal axis of fig. 5 is the urban rail line all-day operation time period, the vertical axis is the station, and the train number marked by the same line type indicates that the same train bottom is used. The abscissa of fig. 6 is time, the ordinate is the number used at the bottom of the car, and the squares on the same horizontal line indicate the number of cars using the same bottom of the car. Fig. 7 is an up-train departure interval chart obtained by cutting out a portion of an urban rail line operation time period all day long, with the horizontal axis representing the number of trains and the vertical axis representing the interval time. As can be seen from the figures 2, 3, 5 and 6, the vehicle bottom is fully utilized after the relevant constraint of the vehicle bottom application is taken into consideration, and the problem of insufficient vehicle bottom caused by the increase of the number of the passenger flow sudden increase vehicles is solved. It can be seen from fig. 4 and 7 that the train operation interval tends to be regular, the operation requirement is met, and the waiting time of passengers is reduced.
In summary, the embodiment of the invention realizes automatic compilation of the train operation diagram based on vehicle bottom multiplexing, and has the following advantages:
1. the automatically compiled train operation diagram can meet the constraint conditions of turn-back time, safety interval, train sequence, train bottom application and the like, the feasibility of the operation diagram is ensured, the operation cost is reduced, and the waiting time of passengers is reduced;
2. the automatic compilation of the train operation diagram can greatly improve the compilation efficiency of the operation diagram and can meet the requirement of frequent adjustment of the train operation diagram under the refined management of an urban rail system.
The above description is only for the preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope of the present invention are included in the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.

Claims (4)

1. A train operation diagram automatic compilation method based on vehicle bottom multiplexing is characterized by comprising the following steps:
configuring basic parameters required for compiling a train operation diagram;
configuring a train running scheme according to the basic parameters;
establishing a train operation diagram optimization model according to the basic parameters and the train operation scheme, wherein the train operation diagram optimization model comprises the following steps: the train dispatching interval calculation model, the train dispatching moment calculation model, the train sequence constraint model, the train bottom application constraint model, the objective function model and the train circulation plan calculation model;
solving the train operation diagram optimization model by adopting a mixed integer linear programming algorithm to obtain a train operation diagram based on vehicle bottom multiplexing;
there are J stations on city rail road, denoted as SstaJ, and the vehicle section is connected with station 1, and defines an operation direction from station 1 to station J as an up direction and an operation direction from station J to station 1 as a down direction;
the configuration train operation scheme comprises the following steps:
starting and stopping time [ t ] of train operationstart，tend]Divided into K time segments, i.e. T1＝[t0，t1]，T2＝[t1，t2]，…，Tk＝[tK-1，tK]Wherein t is0＝tstart，tK＝tend；
Each time period is required to finish IkIndividual number of cars with total number of cars Itotal＝∑k∈{1，2，...，K}IkThe set of the train numbers is Sservice＝{1，2，…，Itotal}；
The train departure interval calculation model comprises:
every two adjacent trains have departure intervals in each time period, and the departure interval of the Kth time period is as follows:and h ismin≤Hk≤hmax，Wherein h ismin、hmaxRespectively representing a minimum departure interval and a maximum departure interval;
the train arrival time calculation model comprises:
the arrival and departure time of the ascending train at each platform has the following calculation formula:
wherein,represents the ascending departure time of the train number i at the station j,representing the time of arrival of the train number i at the station j,represents the up-going stop time of the train number i at the station j,representing the uplink running time of the train number i from the station j to the next station j + 1;
the arrival and departure time of the descending train at each platform has the following calculation formula:
wherein,indicating the time of departure of the train number i at the station j,indicating the time of arrival of the train at station j at train number i,indicating the down stop time of train number i at station j,representing the downlink running time of the train number i from the station j to the next station j-1;
the basic parameters comprise: basic parameters needed for compiling a train running diagram are configured according to an actual train line and an operation condition, wherein the basic parameters comprise line information, station running time, in-station parking time, train operation starting and stopping time, first and last bus time points, a minimum departure interval, a maximum departure interval, a minimum turn-back time and a maximum turn-back time;
and acquiring departure time of the first and last trains according to the time points of the first and last trains, wherein the calculation formula is as follows:
departure time of the first train at the starting station:
departure time of the last train at the starting station:
wherein, the arrival and departure time of two adjacent trains meets the condition
If the train turns back after the station J is the station, the turn-back time of the train at the station JThe following conditions are satisfied:
wherein,andrespectively representing the minimum turn-back time and the maximum turn-back time of the train at a station J;
wherein, γi，i′Whether the train number i and the train number i' use the same train bottom is represented:
the objective function model comprises:
minimizing an objective function Wherein, w1、w2And w3Is a weight, fobj，1，nomNominal value of departure interval deviation, f, provided for mode of serviceobj，2，nomFor using the number of trains at the same bottom, fobj，3，nomA penalty value for interval changes;
in the objective function fobj，1Is a portion that minimizes departure interval deviation from service mode provision, which is expressed as:
in the objective function fobj，2The number of cars using the same car bottom in the operation stage is maximized, which is expressed as:
in the objective function fobj，3Is a part that limits the variation of the interval of the adjacent train, which is represented by the absolute value of the difference between the current interval and the average value of the interval of the adjacent train, that is:
wherein d isi，j-di-1，jThe running interval of the train number i and the train number i-1,
is the average value of the running interval of the train number i and the adjacent train number i which is the front1Number of cars and rear i2The number of the vehicle;
the arrival time of two adjacent trains at the station 1 needs to satisfy the following turn-back condition:
wherein,the time of departure of the train number i +1 at the station 1,the time when the train number i 'reaches the ascending platform of the station 1 is determined, if the train number i and the train number i' are connected with each other, gamma isi，i′If 1, the vehicle is requiredAfter time i' reaches the station 1 ascending platform, the time i +1 stopping at the station 1 descending platform can be sent out from the platform, that is, the train is
The method for solving the train operation diagram optimization model by adopting the mixed integer linear programming algorithm to obtain the train operation diagram based on the vehicle bottom multiplexing comprises the following steps:
step 1, linearizing the real number multiplied by an integer term: in the retrace condition constraint satisfied by the arrival time of the two adjacent trains at the station 1Partially linearized byWherein,
zi，i′≤tKγi，i′，zi，i′≥t0γi，i′，
step 2, pair fobj，1The absolute value part of (a) is linearized: the concrete implementation method is
Wherein,
step 3, pair fobj，3The absolute value part of (a) is linearized: the concrete implementation method isWherein,
and 4, outputting an optimal solution: inputting configuration parameters, solving the train working diagram optimization model by adopting a mixed integer linear programming algorithm, and drawing a train working diagram in an auxiliary manner;
wherein d isi-1，jRepresents the upstream departure time of the previous vehicle in the train number i at the station j,indicating the time of arrival of the train number i at station 1.
2. The train bottom reuse-based train working diagram automatic compilation method according to claim 1, wherein the train sequence constraint model comprises:
based on the basic principle of train sequence constraint, the two-train departure time of the train at the station 1 and the train at the station J meets the following constraint:
3. the train bottom reuse-based train working diagram automatic compilation method according to claim 1, wherein the train circulation plan calculation model comprises:
the train circulation plan calculation method comprises the following steps:
αiindicates whether the train number i is issued by a vehicle segment:
βiindicates whether the train number i returns to the vehicle section:
。
4. the train bottom reuse-based train working diagram automatic compilation method according to claim 3, characterized in that the train circulation plan calculation meets the following constraint conditions:
whether the train number i is sent out by the train section depends on whether the train number i' and the previous train number use the same train bottom, namely the relationship is satisfied:
whether the train number i returns to the train section depends on whether the train number i' exists behind and whether the train number i uses the same train bottom, namely the relationship is satisfied:
when the train number i and the train number i 'use the same train bottom, the train number i does not return to the train section, and the train number i' is not sent out by the train section, namely alphai′、βiAnd gammai，i′The following relationship is satisfied:
αi′+βi≤2+M(1-γi，i′)，
αi′+βi≥2-M(1-γi，i′)；
if the train turns back after the station 1 is the station, the arrival time of two trains i, i '(i is the front train and i' is the rear train) before and after the station 1 should satisfy the following relationship:
wherein,andthe minimum and maximum turnaround times of the train at the station 1 are respectively indicated.
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