CN113408906B - Joint optimization method for high-speed train stop and passenger flow distribution - Google Patents

Joint optimization method for high-speed train stop and passenger flow distribution Download PDF

Info

Publication number
CN113408906B
CN113408906B CN202110690646.7A CN202110690646A CN113408906B CN 113408906 B CN113408906 B CN 113408906B CN 202110690646 A CN202110690646 A CN 202110690646A CN 113408906 B CN113408906 B CN 113408906B
Authority
CN
China
Prior art keywords
solving
stage
model
train
passenger
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Active
Application number
CN202110690646.7A
Other languages
Chinese (zh)
Other versions
CN113408906A (en
Inventor
张玉利
周超越
梁熙栋
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Beijing Institute of Technology BIT
Original Assignee
Beijing Institute of Technology BIT
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Beijing Institute of Technology BIT filed Critical Beijing Institute of Technology BIT
Priority to CN202110690646.7A priority Critical patent/CN113408906B/en
Publication of CN113408906A publication Critical patent/CN113408906A/en
Application granted granted Critical
Publication of CN113408906B publication Critical patent/CN113408906B/en
Active legal-status Critical Current
Anticipated expiration legal-status Critical

Links

Images

Classifications

    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
    • G06Q10/00—Administration; Management
    • G06Q10/06—Resources, workflows, human or project management; Enterprise or organisation planning; Enterprise or organisation modelling
    • G06Q10/063—Operations research, analysis or management
    • G06Q10/0631—Resource planning, allocation, distributing or scheduling for enterprises or organisations
    • G06Q10/06315—Needs-based resource requirements planning or analysis
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
    • G06Q10/00—Administration; Management
    • G06Q10/04—Forecasting or optimisation specially adapted for administrative or management purposes, e.g. linear programming or "cutting stock problem"
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
    • G06Q10/00—Administration; Management
    • G06Q10/06—Resources, workflows, human or project management; Enterprise or organisation planning; Enterprise or organisation modelling
    • G06Q10/063—Operations research, analysis or management
    • G06Q10/0637—Strategic management or analysis, e.g. setting a goal or target of an organisation; Planning actions based on goals; Analysis or evaluation of effectiveness of goals
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
    • G06Q50/00—Information and communication technology [ICT] specially adapted for implementation of business processes of specific business sectors, e.g. utilities or tourism
    • G06Q50/40—Business processes related to the transportation industry
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
    • Y02T10/00—Road transport of goods or passengers
    • Y02T10/10—Internal combustion engine [ICE] based vehicles
    • Y02T10/40—Engine management systems

Landscapes

  • Business, Economics & Management (AREA)
  • Human Resources & Organizations (AREA)
  • Engineering & Computer Science (AREA)
  • Economics (AREA)
  • Strategic Management (AREA)
  • Entrepreneurship & Innovation (AREA)
  • Tourism & Hospitality (AREA)
  • Theoretical Computer Science (AREA)
  • General Physics & Mathematics (AREA)
  • Marketing (AREA)
  • General Business, Economics & Management (AREA)
  • Physics & Mathematics (AREA)
  • Educational Administration (AREA)
  • Quality & Reliability (AREA)
  • Operations Research (AREA)
  • Game Theory and Decision Science (AREA)
  • Development Economics (AREA)
  • Health & Medical Sciences (AREA)
  • General Health & Medical Sciences (AREA)
  • Primary Health Care (AREA)
  • Management, Administration, Business Operations System, And Electronic Commerce (AREA)

Abstract

The invention discloses a joint optimization method for a high-speed train stop scheme and passenger flow distribution, which comprises the following steps: considering uncertainty of the passenger travel demand, describing the passenger travel demand based on an uncertainty set of conditional probability; constructing a two-stage robust optimization model based on the uncertainty set; the train stop scheme is used as a first-stage decision variable, and passenger flow distribution is used as a second-stage decision variable; analyzing and equivalently converting the two-stage robust optimization model, and simplifying an uncertainty set; solving the converted two-stage robust optimization model by adopting a row-column generation algorithm; and solving the subproblems in the row-column generation algorithm by adopting an accurate solving method and an approximate solving method of the KKT condition. The invention considers the uncertainty of the trip demand of the passenger, so that the decision is more in line with the actual situation, and the practicability is stronger; the subproblem approximate solving method can rapidly obtain a high-precision solution and can provide a decision basis for an actual high-speed train stop scheme and passenger ticket distribution.

Description

Joint optimization method for high-speed train stop and passenger flow distribution
Technical Field
The invention belongs to the technical field of high-speed train operation management, relates to a joint optimization method for high-speed train stop and passenger flow distribution, and particularly relates to a joint optimization method for a high-speed train stop scheme and passenger flow distribution considering uncertain travel demands.
Background
The continuous expansion of the scale of the railway network and the continuous increase of the number of passengers also present a great challenge to the operation organization capacity of the railway department, and the railway passenger transport system is used as a comprehensive, complex, dynamic and open large system and comprises a plurality of decision optimization problems. The railway operation decision planning process is generally divided into three levels of strategy, tactics and execution, and a train stop scheme and passenger flow distribution are used as important links of tactical level decision, so that the railway operation efficiency is obviously influenced. The stop scheme is that after the train driving section is determined, the stop station sequence of the train is determined, and passenger flow distribution is to distribute passengers with different OD requirements to each station along the train; the reasonable station stopping scheme can meet the passenger flow requirements among all OD pairs as much as possible, reduce the cost waste caused by unnecessary station stopping and improve the train service level; reasonable passenger flow distribution can increase the seat-boarding rate of the train and improve the passenger service level and the operation income of the railway department. Because the train can only provide transportation service for passengers in the station where the train stops, the arrangement of the train stop scheme has decisive influence on the distribution of train passenger flow; different train stop schemes correspond to different passenger flow distribution results, so that different transportation services are provided for passengers.
The passenger demand is important information influencing a train stop scheme and passenger flow distribution, and in practice, the passenger travel demand is influenced by factors such as weather, festivals and holidays and has strong fluctuation. If the passenger demand is regarded as a deterministic parameter or a random parameter obeying certain probability distribution, the obtained train stop scheme and the passenger flow distribution result are possibly not matched with the passenger travel demand, the transport capacity of the train is wasted, and the seat occupancy rate and the transport income of the high-speed railway are reduced.
At present, research contents related to the stop scheme are various, research scenes are rich, and a system is not formed yet. From different perspectives, there are different model optimization objectives. For example, from the perspective of railway operators, common model optimization objectives include high operation profit, lowest cost, shortest train operation time, minimum total stop times and the like, and for example, Lin, Ku, sun-hui-an and the like, the maximum operation profit of railway departments is taken as an optimization objective to develop research related to stop schemes. The model is constructed from the perspective of passengers, common model optimization targets comprise shortest travel time of the passengers, lowest travel cost and the like, such as Zhengli and the like and Bingmikai and the like, and the minimum total time consumed by the passengers in traveling is taken as the optimization target to carry out research related to the stop scheme. Some scholars consider the joint optimization of the stop scheme and other decisions, such as passenger flow distribution, train schedules, train operation frequency, and the like. The Cacchiani and the like, Chang and the like and Qi and the like simultaneously consider train stop decision and passenger flow distribution decision, the model gives the number of passengers to be carried by trains between different OD pairs through the passenger flow distribution decision, and the decision of train operation frequency is also included in the stop scheme model established by Chang and the like (2000).
Through the retrieval and analysis of past documents, most scholars often add scheduling decisions related to the scholars when the scholars conduct the stop scheme research, such as train schedules, passenger flow distribution and the like. Most studies consider the passenger flow as a deterministic parameter, and some studies assume that the passenger flow is a random parameter subject to a known distribution, and few documents model the parking optimization-related problem from the viewpoint of robust optimization, so that the robust optimization problem related to the parking scheme has a large study space. The invention considers the uncertainty of railway passenger flow demand and researches the joint optimization problem of train stop scheme and passenger flow distribution.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a high-speed train stop scheme and passenger flow distribution combined optimization method considering uncertain travel demands.
The invention discloses a joint optimization method for a high-speed train stop scheme and passenger flow distribution, which comprises the following steps:
considering uncertainty of the passenger travel demand, describing the passenger travel demand based on an uncertainty set of conditional probability;
constructing a two-stage robust optimization model based on the uncertainty set; the train stop scheme is used as a first-stage decision variable, and passenger flow distribution is used as a second-stage decision variable;
analyzing and equivalently transforming the two-stage robust optimization model, and simplifying an uncertainty set;
solving the converted two-stage robust optimization model by adopting a row-column generation algorithm; and solving the subproblems in the row-column generation algorithm by adopting an accurate solving method and an approximate solving method of the KKT condition.
As a further improvement of the invention, in describing the travel demand of passengers based on the uncertainty set of the conditional probability, the OD passenger flow d of the high-speed railway is considered to be p m Is in the uncertainty set D m 。
As a further improvement of the invention, a two-stage robust optimization model is constructed with the goal of minimizing the total time lost to train stopping as a first stage model and the goal of minimizing the number of unmet passenger demands as a second stage model.
As a further improvement of the present invention, in the first stage model, the model constraints include: the minimum number of services per stop, the minimum number of stops per train and all trains have the same starting and ending stations.
As a further improvement of the present invention, in the second stage model, the model constraints include: the traffic assigned to any OD pair should not exceed the actual traffic demand of that OD pair, the traffic assignment decision variable y kij Whether it is 0 depends on whether the train stops at i and j stops at the same time and the number of passengers loaded by the train should not exceed the number of train determiners.
As a further improvement of the method, the equivalent form of the model is obtained by solving the dual problem of the second-stage model, and the model is analyzed and the uncertainty set is simplified.
As a further improvement of the method, a row-column generation algorithm is adopted to solve the converted two-stage robust optimization model, and the method comprises the following steps:
initializing a lower bound LB and an upper bound UB;
solving the main problem of the model and obtaining the optimal solution of the problem, and updating a lower bound LB;
solving the subproblem and updating the upper bound UB;
if UB-LB < is in the same form, the epsilon is a constant; and obtaining an optimal solution, otherwise, adding constraints to the limited main problem, and solving the model main problem and the sub-problems again.
As a further improvement of the invention, the method for solving the subproblems in the row-column generation algorithm by adopting the precise solution method and the approximate solution method of the KKT condition comprises the following steps:
the precise solving method based on the KKT condition comprises the following steps: to solve the problems
Figure BDA0003126571480000031
By obtaining the optimal solution y by the KKT condition of the problem Z (x, d) * (ii) a Providing a KKT condition, because the KKT condition comprises nonlinear constraint, linearizing the KKT condition by introducing a variable of 0-1 and utilizing a large M method, and finally changing the subproblem into a solvable form;
the approximate solution method comprises the following steps: first, define
Figure BDA0003126571480000032
Then, given x, the problem is solved approximately by the following steps
Figure BDA0003126571480000033
(1) Solving a linear programming problem Z (x, mu);
(2) judging whether the condition is satisfied
Figure BDA0003126571480000034
If yes, directly obtaining an optimal solution; otherwise, the problem is approximately converted into a mixed integer linear programming problem to be solved.
Compared with the prior art, the invention has the beneficial effects that:
the invention researches the joint optimization problem of train stop scheme and passenger flow distribution under the condition of uncertain passenger travel demand, and considers the uncertainty of passenger travel demand, so that the decision is more in line with the actual situation, and the practicability is stronger; compared with the model solving result based on the traditional uncertainty set, the model solving result based on the conditional probability uncertainty set has reduced conservatism; compared with an accurate solving method based on the Countque condition, the approximate solving method has the advantages that the solving time is greatly improved, and the optimal value deviation of the approximate solving method and the accurate solving method is small.
Drawings
Fig. 1 is a flowchart of a joint optimization method for a stop plan and passenger flow allocation of a high-speed train considering uncertain travel demands according to an embodiment of the invention;
fig. 2 is a schematic diagram of the order of occurrence of the decision and the actual passenger flow demand according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all, embodiments of the present invention. All other embodiments, which can be obtained by a person skilled in the art without making any creative effort based on the embodiments in the present invention, belong to the protection scope of the present invention.
The invention is described in further detail below with reference to the attached drawing figures:
the invention provides a joint optimization method for a stop scheme and passenger flow distribution of a high-speed train considering uncertain travel demands, which is researched aiming at the stop scheme and the passenger flow distribution of the train, and two important factors are considered in an optimization model: firstly, the travel demand of passengers has random fluctuation and has different travel demand characteristics in different scenes (working days, holidays, cold and summer holidays and the like), so that the uncertainty set based on conditional probability is adopted in the model to describe the passenger demand; secondly, passengers can get on or off the train at the stop station where the train stops, and the arrangement of the train stop scheme has a decisive influence on the passenger flow distribution scheme, so the invention jointly optimizes the stop scheme and the passenger flow distribution.
The train stop scheme has a significant impact on the distribution of passenger flow because passengers can only get on or off at the stop of the train, and only the OD pairs combined by the collection of stop at which the train stops can be assigned passenger flow. Based on the above features, as shown in fig. 2, the present invention divides the decision into two stages: in the first stage, the stop scheme is decided, and at the moment, a decision maker does not know the specific value of the passenger flow demand and only knows the mean value and the standard deviation of the passenger flow; then the real value of the passenger flow is observed by a decision maker; and in the second stage, passenger flow distribution decision is correspondingly carried out according to the actual passenger flow value.
Specifically, the method comprises the following steps:
as shown in fig. 1, the invention provides a joint optimization method for a stop scheme and passenger flow allocation of a high-speed train considering uncertain travel demands, which comprises the following steps:
step 1, considering uncertainty of passenger travel demands, and describing the passenger travel demands based on an uncertainty set of conditional probability; wherein,
in the process of describing the travel demand of passengers based on the uncertainty set of the conditional probability, the OD passenger flow d of the high-speed railway is considered to be p m Is in the uncertainty set D m Where M is equal to {1, … M }.
Based onThis, set uncertainty D m Is defined as:
Figure BDA0003126571480000051
step 2, performing combined optimization on the train stop scheme and passenger flow distribution based on the uncertainty set, and constructing a two-stage robust optimization model; the train stop scheme is used as a first-stage decision variable, and passenger flow distribution is used as a second-stage decision variable;
the two-stage robust optimization model is constructed, and specifically comprises the following steps:
1) describing the parameters in the optimization model:
Figure BDA0003126571480000052
2) the total time of the train lost due to station stopping is minimized to serve as the target of a first-stage model, the quantity of the unmet passenger demands is minimized to serve as the target of a second stage, and a two-stage robust optimization model is constructed;
the model objectives are as follows:
Figure BDA0003126571480000061
Figure BDA0003126571480000062
Figure BDA0003126571480000063
Figure BDA0003126571480000064
Figure BDA0003126571480000065
wherein Z is m (x,d m ) The expression is as follows:
Figure BDA0003126571480000066
Figure BDA0003126571480000067
Figure BDA0003126571480000068
Figure BDA0003126571480000069
Figure BDA00031265714800000610
Figure BDA00031265714800000611
(1) formula is the objective function of the first stage; in the first stage model, constraint (2) gives the minimum number of stop trains allowed for each stop, constraint (3) gives the minimum number of stop stations for each train, and constraint (4) shows that all trains have the same starting and ending stations.
(6) The formula is an objective function of the second stage; in the second stage model, constraints (7) indicate that the traffic assigned to any OD pair should not exceed the actual traffic demand for that OD pair, and constraints (8) and (9) indicate that the traffic assignment decision variable y kij Whether it is 0 depends on whether the train stops at i and j stops at the same time, and the constraint (10) indicates that the number of passengers loaded by the train should not exceed the number of train stops.
Step 3, analyzing and equivalently transforming the two-stage robust optimization model, and simplifying an uncertainty set; wherein,
the method obtains the equivalent form of the second-stage model by solving the dual problem of the second-stage model, analyzes the model and simplifies the uncertainty set; the method specifically comprises the following steps:
for convenience of representation, subscript m is omitted in this section; for constraints (8), (9) of the second stage problem, the invention introduces E k (x) As follows, E k (x)＝{(i,j):x ki x kj ＝1,1≤i<j ≦ S | }, then Z (x, d) may be restated as:
Figure BDA0003126571480000071
Figure BDA0003126571480000072
Figure BDA0003126571480000073
Figure BDA0003126571480000074
Figure BDA0003126571480000075
to simplify the uncertainty set, first the invention will max d∈D Z (x, d) translates into a maximization problem. Given x and d, Z (x, d) is a minimized linear programming problem; its dual problems are:
Figure BDA0003126571480000076
Figure BDA0003126571480000077
Figure BDA0003126571480000078
Figure BDA0003126571480000079
depending on the resulting dual problem, the problem max is given by x d∈D Z (x, d) can be restated as the maximization problem (P).
Figure BDA00031265714800000710
Figure BDA00031265714800000711
Figure BDA00031265714800000712
Figure BDA00031265714800000713
Figure BDA00031265714800000714
Figure BDA0003126571480000081
The invention analyzes the coefficient of the passenger flow demand d in the model (P), and the passenger flow d of the known high-speed rail is more than or equal to 0. In the case of a given value of β, α ij The optimum value of (d) can be given by the following formula:
Figure BDA0003126571480000082
therefore, d ij Coefficient of (1-a) ij ≥0。
The invention defines an uncertainty set
Figure BDA0003126571480000083
The following were used:
Figure BDA0003126571480000084
there is the following theorem.
Theorem 1.
Figure BDA0003126571480000085
And (3) proving that: the optimal solution of (P) is expressed as (d) * ,α * ,β * ) Assuming the presence of d ij Optimum value of (2)
Figure BDA0003126571480000086
Fall in [ mu ] ij -δ ij ,μ ij ) Within the range.
Order to
Figure BDA0003126571480000087
Then in accordance with the constraint definition of the passenger flow demand,
Figure BDA0003126571480000088
is also feasible, denoted as feasible solution (d) ′ ,α * ,β * )。
Note book
Figure BDA0003126571480000089
Known as d ij Coefficient of (1-a) ij Greater than or equal to 0, then F (d) ′ ,α * ,β * )＞F(d * ,α * ,β * ) Therefore do not exist
Figure BDA00031265714800000810
Fall in [ mu ] ij -δ ij ,μ ij ) Within the range of (A) and (B),
so that the constraint d can be ij ∈[μ ij -δ ij ,μ ij +δ ij ]Decrease to d ij ∈[μ ij ,μ ij +δ ij ]Thus, the uncertainty set becomes:
Figure BDA00031265714800000811
get z ij ＝d ij -μ ij Then z is ij ∈[0,δ ij ]D is mixing ij ＝z ij +μ ij Carry-in constraint sigma 1≤i<j≤|S| (d ij -μ ij ) Sigma is not more than b 1≤i<j≤|S| (d ij -μ ij )＝∑ 1≤i<j≤|S| z ij B is not more than b, thus, D 1 Equivalent to the following form:
Figure BDA00031265714800000812
therefore, the first and second electrodes are formed on the substrate,
Figure BDA00031265714800000813
step 4, solving the converted two-stage robust optimization model by adopting a row-column generation algorithm; solving subproblems in a row-column generation algorithm by adopting an accurate solving method and an approximate solving method of a Kurush-Kuhn-Tucker (KKT) condition;
specifically, the method comprises the following steps:
the line and column generation algorithm is a secant Plane method, and compared with a Benders secant Plane (BCP) method, the line and column generation algorithm has higher execution speed in solving a two-stage robust problem. In order to simplify the model, an auxiliary variable z is introduced into the original two-stage robust optimization problem m Then the original model can be restated as:
Figure BDA0003126571480000091
Figure BDA0003126571480000092
Figure BDA0003126571480000093
Figure BDA0003126571480000094
Figure BDA0003126571480000095
Figure BDA0003126571480000096
for each d m ∈D m Constraint { z m ≥Z m (x,d m ) Can be rewritten as:
Figure BDA0003126571480000097
Figure BDA0003126571480000098
Figure BDA0003126571480000099
Figure BDA00031265714800000910
Figure BDA00031265714800000911
Figure BDA00031265714800000912
wherein, y mu A newly created decision variable; thus, the two-stage robust optimization model is equivalently transformed into a large-scale mixed integer programming problem. Set D due to uncertainty m Being polyhedral, the present invention cannot solve (P1) for the optimal value by enumerating all uncertainty scenarios in the uncertainty set, but partial enumeration of the uncertainty set scenarios may provide an effective relaxation (and thus a lower bound) of the original two-stage robust optimization problem. A better lower bound can be obtained by extending the enumeration by adding valid scenes step by step. Line and column generation algorithm by solving problems
Figure BDA00031265714800000913
To identify important scenes and to add to the subset of uncertainty sets.
The specific row and column generation algorithm comprises the following steps:
[1] initialization of LB ═ infinity, UB ∞, t ∞ 0
[2] Solving the following main problem:
Figure BDA0003126571480000101
s.t.(2)-(5),
Figure BDA0003126571480000102
Figure BDA0003126571480000103
Figure BDA0003126571480000104
Figure BDA0003126571480000105
Figure BDA0003126571480000106
Figure BDA0003126571480000107
Figure BDA0003126571480000108
solving the problem (MP) and obtaining an optimal solution to the problem
Figure BDA0003126571480000109
Updating the lower bound
Figure BDA00031265714800001010
[3]Solving the subproblems: x is to be * Bringing in and solving problems
Figure BDA00031265714800001011
Obtaining the optimal solution d of passenger flow m,t+1* Update the upper bound
Figure BDA00031265714800001012
[4]Calculating gap if UB-LB<E (e is a relatively small number and can be taken as 10 -6 ) Then return to
Figure BDA00031265714800001013
And ending; otherwise t +1, for all M ∈ {1, …, M }, proceed as follows and return to step 2:
a) if it is not
Figure BDA00031265714800001014
Creating a variable y m,t+1 Add the following constraints to the MP:
Figure BDA00031265714800001015
Figure BDA00031265714800001016
Figure BDA00031265714800001017
Figure BDA00031265714800001018
Figure BDA00031265714800001019
updating t to t +1 and returning to step 2
b) If it is not
Figure BDA00031265714800001020
Creating a variable y m,t+1 Add the following constraints to the MP:
(21)-(25),
Figure BDA0003126571480000111
updating t to t +1 and returning to step 2
The third step of the row and column generation algorithm is to solve the problem
Figure BDA0003126571480000112
In order to solve the problem accurately, the invention provides a KKT strip-based methodThe precise solving method of the element is designed with an approximate solving method in order to accelerate the solving speed. For convenience of representation, the subscript m is omitted hereinafter.
The precise solving method based on the KKT condition is as follows:
to solve the problem accurately
Figure BDA0003126571480000113
The present invention uses the classical KKT condition to handle a set of polyhedral uncertainties in the problem. For convex optimization problems, the KKT condition is a sufficient necessary condition to determine whether a solution of the optimization problem is an optimal solution. Due to the problems
Figure BDA0003126571480000114
The inner layer problem of (2) is a convex optimization problem, so the invention can obtain the optimal solution y through the KKT condition of the problem Z (x, d) * . Given that α, β are dual variables of the second stage problem, the KKT condition is as follows:
(13)-(16),(18)-(20),#(26)
Figure BDA0003126571480000115
Figure BDA0003126571480000116
Figure BDA0003126571480000117
thus, problems arise
Figure BDA0003126571480000118
Equivalent to the following problems:
Figure BDA0003126571480000119
s.t.(13)-(16),(18)-(20),(27)-(29),#(31)
Figure BDA00031265714800001110
Figure BDA00031265714800001111
aiming at the nonlinear constraint in the problem, the invention can linearize the nonlinear constraint by introducing a variable from 0 to 1 and utilizing a large M method. For constraints (27), (28), (29), a 0-1 variable v is introduced, respectively ij ,π kt ,q kij Then the constraint can be restated as:
Figure BDA00031265714800001112
Figure BDA0003126571480000121
Figure BDA0003126571480000122
Figure BDA0003126571480000123
Figure BDA0003126571480000124
Figure BDA0003126571480000125
where U is a maximum. From the previous analysis, it can be seen that the maximum value of U in constraint (34) is taken as 1, and the maximum value of U in constraint (35) is taken as μ ij +δ ij The maximum value of U in other constraints can be obtained by analysis in the same way, soThe constraints can be restated as:
Figure BDA0003126571480000126
Figure BDA0003126571480000127
Figure BDA0003126571480000128
Figure BDA0003126571480000129
Figure BDA00031265714800001210
Figure BDA00031265714800001211
therefore, problems arise
Figure BDA00031265714800001212
Converted to a mixed integer linear programming problem that can be solved by a solver.
Approximate solution method
The precise solving method based on the KKT condition is low in solving speed, so that an approximate solving method is designed. According to theorem 1, problems
Figure BDA00031265714800001213
Can be expressed in the following form:
Figure BDA00031265714800001214
Figure BDA00031265714800001215
Figure BDA00031265714800001216
Figure BDA00031265714800001217
Figure BDA0003126571480000131
Figure BDA0003126571480000132
given x and z, consider the following problem.
Figure BDA0003126571480000133
Figure BDA0003126571480000134
Figure BDA0003126571480000135
Figure BDA0003126571480000136
For a given z, let (α (z), β (z)) be the optimal solution for the problem (P1), then (α (0), β (0)) be the optimal solution for the problem (P2) when z is 0. Order to
Figure BDA0003126571480000137
The inventionThere are the following propositions.
Proposition 1, if
Figure BDA0003126571480000138
Then
Figure BDA0003126571480000139
Is the optimal solution of the problem (SP) when
Figure BDA00031265714800001310
When the temperature of the water is higher than the set temperature,
Figure BDA00031265714800001311
when in use
Figure BDA00031265714800001312
In this case, the following conditions are satisfied:
Figure BDA00031265714800001313
and (3) proving that: firstly, the invention proves that pi (x) is less than or equal to Z (x, mu) + b. The present invention is based on the optimality of (. alpha. (0), beta. (0)), for any z
Figure BDA00031265714800001314
For any z ≧ 0 because α ij (z) is not less than 0, the invention has
Figure BDA00031265714800001315
Thus, it is possible to prevent the occurrence of,
Figure BDA00031265714800001316
Figure BDA0003126571480000141
next, the present invention proves
Figure BDA0003126571480000142
Is a feasible solution to the problem (SP) and the objective function value is Z (x, μ) + b. Due to the fact that
Figure BDA0003126571480000143
And optimality of (. alpha. (0), and. beta. (0), it is found that
Figure BDA0003126571480000144
Is a viable solution to the problem (SP). Which corresponds to an objective function value of
Figure BDA0003126571480000145
In light of the above proposition, given x, the present invention can approximate the Solution Problem (SP) by the following steps.
(1) Solving a linear programming problem Z (x, mu);
(2) judging whether the condition is satisfied
Figure BDA0003126571480000146
If so, directly obtaining an optimal solution; otherwise, the problem (SP) is approximately converted into a mixed integer linear programming problem to be solved.
The approximate transformation of the (SP) problem is as follows. In case of satisfying the constraint, z in the (SP) problem ij Will be taken at the interval boundary, so the present invention introduces the variable u ij E {0,1}, then z ij ＝u ij ×δ ij (SP) becomes approximately (SP1) form:
Figure BDA0003126571480000147
Figure BDA0003126571480000148
Figure BDA0003126571480000149
Figure BDA00031265714800001410
Figure BDA00031265714800001411
Figure BDA00031265714800001412
due to the optimal solution of alpha in (SP1)
Figure BDA00031265714800001413
The invention uses the following formula to convert the bilinear term w ij ＝α ij u ij Linearization
w ij ≥α ij +u ij -1,w ij ≥0.
Thus, the (SP) problem can be approximately translated into a mixed integer programming problem as follows
Figure BDA00031265714800001414
Figure BDA0003126571480000151
Figure BDA0003126571480000152
Figure BDA0003126571480000153
Figure BDA0003126571480000154
Figure BDA0003126571480000155
Figure BDA0003126571480000156
Figure BDA0003126571480000157
The invention has the advantages that:
the invention researches the joint optimization problem of train stop scheme and passenger flow distribution under the condition of uncertain passenger travel demand, and considers the uncertainty of passenger travel demand, so that the decision is more in line with the actual situation, and the practicability is stronger; compared with the model solving result based on the traditional uncertainty set, the model solving result based on the conditional probability uncertainty set has reduced conservatism; compared with an accurate solving method based on the Countack condition, the approximate solving method has the advantages that the solving time is greatly improved, and the optimal value deviation of the approximate solving method and the accurate solving method is small.
The above is only a preferred embodiment of the present invention, and is not intended to limit the present invention, and various modifications and changes will occur to those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (5)

1. A joint optimization method for a high-speed train stop scheme and passenger flow distribution is characterized by comprising the following steps:
considering uncertainty of passenger travel demand, describing the passenger travel demand based on uncertainty set of conditional probability, and considering that OD passenger flow d of the high-speed railway is p m Is in the uncertainty set D m ；
Constructing a two-stage robust optimization model based on the uncertainty set; the train stop scheme is used as a first-stage decision variable, and passenger flow distribution is used as a second-stage decision variable; constructing a two-stage robust optimization model by taking the total time of the train lost due to station stopping as a target of a first-stage model and the quantity of the unsatisfied passenger demands as a target of a second stage;
analyzing and equivalently transforming the two-stage robust optimization model, and simplifying an uncertainty set;
solving the converted two-stage robust optimization model by adopting a row-column generation algorithm; solving sub-problems in a row-column generation algorithm by adopting an accurate solving method and an approximate solving method of a KKT condition; solving the converted two-stage robust optimization model by adopting a row-column generation algorithm, wherein the method comprises the following steps: initializing a lower bound LB and an upper bound UB; solving the main problem of the model and obtaining the optimal solution of the problem, and updating a lower bound LB; solving the subproblem and updating the upper bound UB; if UB-LB < is in the same form, the epsilon is a constant; and obtaining an optimal solution, otherwise, adding constraints to the limited main problem, and solving the model main problem and the sub-problems again.
2. The joint optimization method of claim 1, wherein in the first stage model, the model constraints comprise: the minimum number of services per stop, the minimum number of stops per train and all trains have the same starting and ending stations.
3. The joint optimization method of claim 1, wherein in the second stage model, the model constraints comprise: the traffic assigned to any OD pair should not exceed the actual traffic demand of that OD pair, the traffic assignment decision variable y kij Whether it is 0 depends on whether the train stops at i and j stops at the same time and the number of passengers loaded by the train should not exceed the number of train determiners.
4. The joint optimization method of claim 1, wherein the equivalence is obtained by solving a dual problem of the second stage model, analyzing the model and simplifying the uncertainty set.
5. The joint optimization method of claim 1, wherein solving the sub-problem in the row-column generation algorithm using the exact solution method and the approximate solution method of the KKT condition comprises:
the precise solving method based on the KKT condition comprises the following steps: to solve the problems
Figure FDA0003784001470000011
By obtaining the optimal solution y by the KKT condition of the problem Z (x, d) * (ii) a Providing a KKT condition, because the KKT condition comprises nonlinear constraint, linearizing the KKT condition by introducing a variable of 0-1 and utilizing a large M method, and finally changing the subproblem into a solvable form;
the approximate solution method comprises the following steps: first, define
Figure FDA0003784001470000012
Then, given x, the problem is solved approximately by the following steps
Figure FDA0003784001470000021
(1) Solving a linear programming problem Z (x, mu);
(2) judging whether the condition is satisfied
Figure FDA0003784001470000022
If yes, directly obtaining an optimal solution; otherwise, the problem is approximately converted into a mixed integer linear programming problem to be solved.
CN202110690646.7A 2021-06-22 2021-06-22 Joint optimization method for high-speed train stop and passenger flow distribution Active CN113408906B (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN202110690646.7A CN113408906B (en) 2021-06-22 2021-06-22 Joint optimization method for high-speed train stop and passenger flow distribution

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN202110690646.7A CN113408906B (en) 2021-06-22 2021-06-22 Joint optimization method for high-speed train stop and passenger flow distribution

Publications (2)

Publication Number Publication Date
CN113408906A CN113408906A (en) 2021-09-17
CN113408906B true CN113408906B (en) 2022-09-23

Family

ID=77682358

Family Applications (1)

Application Number Title Priority Date Filing Date
CN202110690646.7A Active CN113408906B (en) 2021-06-22 2021-06-22 Joint optimization method for high-speed train stop and passenger flow distribution

Country Status (1)

Country Link
CN (1) CN113408906B (en)

Families Citing this family (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN114596008A (en) * 2022-05-09 2022-06-07 支付宝(杭州)信息技术有限公司 Robust optimization method for resource allocation decision considering prediction uncertainty

Family Cites Families (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN107330539A (en) * 2017-05-18 2017-11-07 北京交通大学 Train schedule chart and the scheme cooperative optimization method that stops based on bus traveler assignment
US10636293B2 (en) * 2017-06-07 2020-04-28 International Business Machines Corporation Uncertainty modeling in traffic demand prediction
CN110782079B (en) * 2019-10-18 2022-07-05 西南交通大学 Method for adjusting energy-saving high-speed railway train running chart based on stop scheme
CN112231796B (en) * 2020-09-15 2024-06-07 北京交通大学 Periodic train stop scheme optimization modeling method considering passenger space-time travel distribution
CN112329347B (en) * 2020-11-06 2022-09-02 西南交通大学 Optimization method for high-speed railway train stop scheme

Also Published As

Publication number Publication date
CN113408906A (en) 2021-09-17

Similar Documents

Publication Publication Date Title
Wang et al. Economic and environmental evaluations in the two-echelon collaborative multiple centers vehicle routing optimization
Wang et al. Two-echelon logistics distribution region partitioning problem based on a hybrid particle swarm optimization–genetic algorithm
Li et al. Reducing CO2 emissions through pricing, planning, and subsidizing rail freight
Gong et al. Train timetabling with dynamic and random passenger demand: A stochastic optimization method
Ma et al. A model for the stop planning and timetables of customized buses
Zhang et al. Integrated line planning and train timetabling through price-based cross-resolution feedback mechanism
Yaghini et al. Multicommodity network design problem in rail freight transportation planning
Kang et al. Last train station-skipping, transfer-accessible and energy-efficient scheduling in subway networks
Ghoseiri et al. A hybrid genetic algorithm for multi-depot homogenous locomotive assignment with time windows
Yaghini et al. Solving train formation problem using simulated annealing algorithm in a simplex framework
CN113408906B (en) Joint optimization method for high-speed train stop and passenger flow distribution
Nie et al. Modeling and solving the last-shift period train scheduling problem in subway networks
Lin et al. An implicit enumeration algorithm for the passenger service planning problem: Application to the Taiwan Railways Administration line
Hou et al. A hybrid metaheuristic algorithm for the heterogeneous school bus routing problem and a real case study
Mo et al. The vehicle routing problem with underground logistics: Formulation and algorithm
Huang et al. Column-and-constraint-generation-based approach to a robust reverse logistic network design for bike sharing
Fu et al. Train Stop Scheduling in a High‐Speed Rail Network by Utilizing a Two‐Stage Approach
Li et al. The effects of the tractor and semitrailer routing problem on mitigation of carbon dioxide emissions
Tang et al. Research on Hub‐and‐Spoke Transportation Network of China Railway Express
Li et al. Joint operating revenue and passenger travel cost optimization in urban rail transit
CN113469451B (en) Customized bus route generation method based on heuristic algorithm
Xiong et al. Split demand one-to-one pickup and delivery problems with the shortest-path transport along real-life paths
Hong et al. Designing sustainable logistics networks for classified municipal solid wastes collection and transferring with multi-compartment vehicles
Sanabria-Rey et al. Solving last-mile deliveries for dairy products using a biased randomization-based spreadsheet. a case study
Tayachi et al. Optimising green vehicle routing problem-a real case study

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
GR01 Patent grant
GR01 Patent grant