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 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.

2. The joint optimization method of claim 1, wherein in describing passenger travel demands based on the uncertainty set of conditional probabilities, the OD passenger flow d of the high-speed railway is considered to be p_mIs in the uncertainty set D_m。

3. The joint optimization method of claim 1, wherein a two-stage robust optimization model is constructed with the goal of minimizing the total time lost to train stops as a first stage model and the goal of minimizing the number of unmet passenger demands as a second stage model.

4. The joint optimization method of claim 3, 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.

5. The joint optimization method of claim 3, 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_kijWhether 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.

6. 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.

7. The joint optimization method of claim 1, wherein solving the transformed two-stage robust optimization model using a row-column generation algorithm comprises:

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.

8. The joint optimization method of claim 7, wherein solving the sub-problems 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 problemsBy 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, defineThen, given x, the problem is solved approximately by the following steps

(1) Solving a linear programming problem Z (x, mu);

(2) judging whether the condition is satisfiedIf yes, directly obtaining an optimal solution; otherwise, the problem is approximately converted into a mixed integer linear programming problem to be solved.

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 a certain probability distribution, the obtained train stop scheme and the passenger flow distribution result are possibly not matched with the passenger travel demand, the transportation capacity of the train is wasted, and the seat occupancy rate and the transportation 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 passengers, lowest travel cost and the like, such as Zhengli and Bingkei, 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_mIs 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_kijWhether 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, the transformed two-stage robust optimization model is solved by adopting a row-column generation algorithm, 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 problemsBy 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, defineThen, given x, the problem is solved approximately by the following steps

(1) Solving a linear programming problem Z (x, mu);

(2) judging whether the condition is satisfiedIf 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 any inventive step based on the embodiments of the present invention, are within the scope of the present invention.

The invention is described in further detail below with reference to the attached drawing figures:

the invention provides a high-speed train stop scheme and passenger flow distribution combined optimization method considering uncertain travel demands, which is researched aiming at the train stop scheme and the passenger flow distribution, 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 present invention provides a joint optimization method for a stop plan and passenger flow allocation of a high-speed train considering an uncertain travel demand, which includes:

step 1, considering uncertainty of passenger travel demands, and describing the passenger travel demands based on an uncertainty set of conditional probability; wherein the content of the first and second substances,

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_mIs in the uncertainty set D_mWhere M is equal to {1, … M }.

Based on this, set uncertainty D_mIs defined as:

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:

2) taking the total time of the train lost due to station stopping as the target of a first-stage model, taking the quantity of the unsatisfied passenger demands as the target of a second stage, and constructing a two-stage robust optimization model;

the model objectives are as follows:

wherein Z is_m(x,d^m) The expression is as follows:

(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 passenger flow assigned to any OD pair should not exceed the actual passenger flow demand of that OD pair, and constraints (8) and (9) indicate that the passenger flow assignment decision variable y_kijWhether 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 content of the first and second substances,

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_kix_kj＝1,1≤i<j ≦ S | }, then Z (x, d) may be restated as:

to simplify the uncertainty set, first the invention will max_d∈DZ (x, d) translates into a maximization problem. Given x and d, Z (x, d) is a minimized linear programming problem; its dual problems are:

depending on the resulting dual problem, the problem max is given by x_d∈DZ (x, d) can be restated as the maximization problem (P).

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_ijThe optimum value of (d) can be given by the following formula:

therefore, d_ijCoefficient of (1-a)_ij≥0。

The invention defines an uncertainty setThe following were used:

there is the following theorem.

Theorem 1.

And (3) proving that: the optimal solution of (P) is expressed as (d)^*,α^*,β^*) Assuming the presence of d_ijOptimum value of (2)Fall in [ mu ]_ij-δ_ij,μ_ij) Within the range.

Order toThen in accordance with the constraint definition of the passenger flow demand,is also feasible, denoted as feasible solution (d)^′,α^*,β^*)。

Note the book

Known as d_ijCoefficient of (1-a)_ijGreater than or equal to 0, then F (d)^′,α^*,β^*)＞F(d^*,α^*,β^*) Therefore do not existFall 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:

get z_ij＝d_ij-μ_ijThen z is_ij∈[0,δ_ij]D is mixing_ij＝z_ij+μ_ijCarry-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_ijB is not more than b, thus, D₁Is equivalent to the followingForm (a):

therefore, the first and second electrodes are formed on the substrate,

step 4, solving the converted two-stage robust optimization model by adopting a row-column generation algorithm; solving the subproblems in the row-column generation algorithm by adopting an accurate solving method and an approximate solving method of Kurush-Kuhn-Tucker (KKT) conditions;

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_mThen the original model can be restated as:

for each d^m∈D_mConstraint { z_m≥Z_m(x,d^m) Can be rewritten as:

wherein, y^muA 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_mBeing 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. By extending the enumeration by adding valid scenes step by step, a better lower bound can be obtained. Line and column generation algorithm by solving problemsTo 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 ═ infinity, and t ═ 0

[2] Solving the following main problem:

s.t.(2)-(5),

solving the problem (MP) and obtaining an optimal solution to the problemUpdating the lower bound

[3]Solving the subproblems: x is to be^*Carry-in and solve problemsObtaining the optimal solution d of passenger flow^m,t+1*Update the upper bound

[4]Calculating gap if UB-LB<E (e is a relatively small number and can be taken as 10^-6) Then return toAnd terminating; otherwise t +1, for all M ∈ {1, …, M }, proceed as follows and return to step 2:

a) if it is notCreating a variable y^m,t+1Add the following constraints to the MP:

updating t to t +1 and returning to step 2

b) If it is notCreating a variable y^m,t+1Add the following constraints to the MP:

(21)-(25),

updating t to t +1 and returning to step 2

The third step of the row and column generation algorithm is to solve the problemIn order to solve the problem accurately, the invention provides an accurate solving method based on the KKT condition, and an approximate solving method is designed 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 accuratelyThe 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 problemsThe inner layer problem of (a) 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)

thus, problems ariseEquivalent to the following problems:

s.t.(13)-(16),(18)-(20),(27)-(29),#(31)

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_kijThen the constraint can be restated as:

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_ijThe maximum values of U in other constraints can be analyzed similarly, so the constraint can be restated as:

therefore, problems ariseConverted 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, questionCan be expressed in the following form:

given x and z, consider the following problem.

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 toThe present invention has the following propositions.

Proposition 1, ifThenIs an optimal solution to the problem (SP) whenWhen the temperature of the water is higher than the set temperature,when in useIn this case, the following conditions are satisfied:

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

For any z ≧ 0 because α_ij(z) is not less than 0, the invention has

Thus, it is possible to prevent the occurrence of,

next, the present invention provesIs a feasible solution to the problem (SP) and the objective function value is Z (x, μ) + b. Due to the fact thatAnd optimality of (. alpha. (0), and. beta. (0), it is found thatIs a viable solution to the problem (SP). Which corresponds to an objective function value of

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 satisfiedIf 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_ijWill be taken at the interval boundary, so the present invention introduces the variable u_ijE {0,1}, then z_ij＝u_ij×δ_ij(SP) becomes approximately (SP1) form:

due to the optimal solution of alpha in (SP1)The inventionThe bilinear term w is expressed by the following formula_ij＝α_iju_ijLinearization

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

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 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.

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.