Bus route contract scheme optimization method and device

Abstract

The invention relates to a bus route contract scheme optimization method and device. According to the method, a three-layer planning model is constructed based on three-party interaction logic of a system planning end, an operator end and a passenger end in a public transit network; solving the three-layer planning model by using a nested generalized Benders decomposition algorithm to obtain a line contract scheme and a corresponding service frequency scheme which minimize the total cost of the system; based on tested bus network data, a numerical experiment is carried out on a line contract scheme and a service frequency scheme to obtain an experiment result, and the method has the advantages that strategy interaction of a system planning end, an operator end and a passenger end can be accurately described, efficient solving is carried out through a nested decomposition algorithm, and the system performance is improved. And finally, outputting an optimal line contract and frequency setting scheme with the total social cost being minimized.

Description

Technical Field

[0001] This invention relates to the field of public transportation system optimization technology, and in particular to a method and apparatus for optimizing bus route contracting schemes. Background Technology

[0002] Public-Private Partnerships (PPPs) have become an important strategy for improving the efficiency and sustainability of urban public transport systems. However, existing research and practice generally suffer from two key limitations: First, most existing methods neglect the differences at the bus route level and fail to explore the potential benefits of selectively outsourcing only a portion of the network. Selective route-level contracting, where the system planners selectively contract out some routes to operators while retaining independent operation of the remaining routes, aligns better with the principle of gradual reform. However, scientifically selecting contracted routes is a complex decision involving interdependent strategic behaviors among the system planners, operators, and passengers, forming a hierarchical game structure. Improperly designed contracting schemes can lead to decreased system performance; for example, over-contracting may compromise service fairness, while under-contracting fails to leverage the efficiency advantages of the private sector.

[0003] Secondly, regarding modeling and solution techniques, while bi-level programming has been widely used to describe the interaction between the system planner and passenger sides, the bi-level structure cannot accurately capture the strategic interactions among the three parties when an operator with independent profit objectives is introduced. A few studies have attempted to use tri-level programming models to describe this problem, but due to the inherent complexity of the models, there is a lack of effective algorithms capable of accurately solving general tri-level optimization problems. Existing methods mostly rely on simplifying the model to a bi-level structure or using heuristic algorithms to find approximate solutions, which cannot guarantee the optimality of the solution, thus affecting the scientific nature of the decision-making process.

[0004] Currently, no effective solution has been proposed to address the problem that existing models are incomplete in their characterization and have insufficient solution capabilities, which prevent them from providing reliable basis for decision-making. Summary of the Invention

[0005] The purpose of this invention is to address the shortcomings of existing technologies by providing an optimization method and apparatus for bus route contracting schemes, thereby solving the technical problem that existing models are incomplete in characterization and have insufficient solution capabilities, resulting in the inability to provide reliable basis for decision-making.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows: This invention provides an optimization method for bus route contracting schemes, comprising: constructing a three-layer planning model based on the three-way interaction logic of the system planner, operator, and passenger in the bus network; applying a nested generalized Benders decomposition algorithm to solve the three-layer planning model to obtain the route contracting scheme and corresponding service frequency scheme that minimizes the total system cost; and conducting numerical experiments on the route contracting scheme and service frequency scheme based on tested bus network data to obtain experimental results.

[0007] Optionally, the three-level programming model includes: an upper-level model with the objective of minimizing the total system cost, and a decision variable that is a binary variable representing whether each bus route is contracted; a middle-level model with the objective of maximizing the net profit of the contracted route operators, and a decision variable that is the service frequency of each contracted route; and a lower-level model that is a bus network equilibrium allocation model used to simulate the passenger flow distribution at a given service frequency.

[0008] Furthermore, optionally, the method also includes: constructing a lower-level model, which includes: constructing a hypergraph model based on the public transport network topology, wherein the nodes of the hypergraph model include bus stops and transfer points, and the road segments include boarding road segments, in-vehicle road segments, alighting road segments, and walking road segments; introducing the concept of hyperpath to model the probabilistic choice behavior of passengers at transfer points, and based on the Wardrop equilibrium principle, describing the conditions for the network to reach an equilibrium state in the form of variational inequalities to obtain the lower-level model.

[0009] Optionally, applying the nested generalized Benders decomposition algorithm to solve the three-level programming model includes: using a logical cutting plane to process discrete decision variables and employing relaxation techniques to accelerate computation; wherein, applying the nested generalized Benders decomposition algorithm includes: combining the middle-level model and the lower-level model into an internal bilevel programming problem and solving it using the Benders decomposition method; introducing auxiliary variables and performing McCormick linearization on the bilinear terms in the objective function to transform the internal bilevel programming problem into a mixed-integer linear programming problem; combining the upper-level model and the solved internal bilevel programming problem into an external bilevel programming problem and solving it using the Benders decomposition method, and gradually approaching the global optimum by iteratively adding optimality cutting planes and feasibility cutting planes.

[0010] Further, optionally, relaxation techniques can be used to accelerate computation, including: constructing a relaxation model of the internal bi-level programming problem, transforming the internal bi-level programming problem into a system optimal allocation problem by relaxing the profit maximization constraint of the middle-level model and the user equilibrium constraint of the lower-level model; and reconstructing the feasible region of passenger flow allocation based on super-segment, expressing the super-path flow as a linear combination of segment flows.

[0011] Optionally, numerical experiments can be conducted on route contracting schemes and service frequency schemes based on test bus network data, including: comparing the computation time and convergence speed of the nested generalized Benders decomposition algorithm before and after the adoption of relaxation techniques under test networks of different sizes and passenger flow demands; and comparing the total system cost, total passenger travel time, and total operator cost indicators of the generated optimized contracting scheme with various benchmark operation schemes.

[0012] This invention provides an optimization device for bus route contracting schemes, comprising: a construction module for constructing a three-layer planning model based on the three-way interaction logic between the system planner, operator, and passenger in a bus network; a calculation module for solving the three-layer planning model using a nested generalized Benders decomposition algorithm to obtain the route contracting scheme and corresponding service frequency scheme that minimizes the total system cost; and a verification module for conducting numerical experiments on the route contracting scheme and service frequency scheme based on test bus network data to obtain experimental results.

[0013] The present invention provides a storage medium storing a computer program, wherein the computer program is configured to execute the above-mentioned optimization method for bus route contracting scheme when running.

[0014] This invention employs the above technical solution, constructing a three-layer planning model based on the three-way interaction logic between the system planner, operator, and passenger in a public transport network. It then applies a nested generalized Benders decomposition algorithm to solve the three-layer planning model, obtaining the route contracting scheme and corresponding service frequency scheme that minimizes the total system cost. Based on tested public transport network data, numerical experiments are conducted on the route contracting scheme and service frequency scheme. The experimental results, compared with existing technologies, demonstrate the following technical advantages: It accurately describes the strategic interactions between the system planner, operator, and passenger, and efficiently solves the problem using a nested decomposition algorithm, ultimately outputting the optimal route contracting and frequency setting scheme that minimizes the total social cost. Attached Figure Description

[0015] Figure 1 This is a flowchart illustrating an optimization method for a bus route contracting scheme according to Embodiment 1 of the present invention. Figure 2 This is a schematic diagram of the technical route in an optimization method for a bus route contracting scheme according to Embodiment 1 of the present invention; Figure 3 This is a schematic diagram of a baseline example of a numerical experiment in an optimization method for a bus route contracting scheme according to Embodiment 1 of the present invention. Figure 4 This is a schematic diagram illustrating the impact of the subsidy coefficient on the outcome indicators in an optimization method for a bus route contracting scheme according to Embodiment 1 of the present invention. Figure 5 This is a schematic diagram illustrating the impact of fares on outcome indicators in an optimization method for a bus route contracting scheme according to Embodiment 1 of the present invention. Figure 6 This is a schematic diagram illustrating the impact of operator cost factors on outcome indicators in an optimization method for a bus route contracting scheme according to Embodiment 1 of the present invention. Figure 7 This is a schematic diagram of an optimization device for a bus route contracting scheme according to Embodiment 2 of the present invention. Detailed Implementation

[0016] To make the objectives, technical solutions, and advantages of this application clearer, the application is described and illustrated below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application. All other embodiments obtained by those skilled in the art based on the embodiments provided in this application without inventive effort are within the scope of protection of this application.

[0017] Obviously, the accompanying drawings described below are merely some examples or embodiments of this application. Those skilled in the art can apply this application to other similar scenarios based on these drawings without any inventive effort. Furthermore, it is understood that although the efforts made in this development process may be complex and lengthy, for those skilled in the art related to the content disclosed in this application, any changes to design, manufacturing, or production based on the technical content disclosed in this application are merely conventional technical means and should not be construed as insufficient disclosure of the content of this application.

[0018] In this application, the reference to "embodiment" means that a specific feature, structure, or characteristic described in connection with an embodiment may be included in at least one embodiment of this application. The appearance of this phrase in various places in the specification does not necessarily refer to the same embodiment, nor is it a separate or alternative embodiment that is mutually exclusive with other embodiments. It will be explicitly and implicitly understood by those skilled in the art that the embodiments described in this application may be combined with other embodiments without conflict.

[0019] Unless otherwise defined, the technical or scientific terms used in this application shall have the ordinary meaning understood by one of ordinary skill in the art to which this application pertains. The terms “a,” “an,” “an,” “the,” and similar words used in this application do not indicate quantity limitation and may indicate singular or plural. The terms “comprising,” “including,” “having,” and any variations thereof used in this application are intended to cover non-exclusive inclusion; for example, a process, method, system, product, or apparatus that includes a series of steps or units (elements) is not limited to the listed steps or units, but may also include steps or units not listed, or may include other steps or units inherent to these processes, methods, products, or apparatus. The terms “connected,” “linked,” “coupled,” and similar words used in this application are not limited to physical or mechanical connections, but may include electrical connections, whether direct or indirect. The terms “multiple” / “several” used in this application refer to two or more. “And / or” describes the relationship between related objects, indicating that three relationships may exist; for example, “A and / or B” can indicate: A alone, A and B simultaneously, or B alone. The character " / " generally indicates that the preceding and following objects are in an "or" relationship. The terms "first," "second," and "third" used in this application are merely to distinguish similar objects and do not represent a specific ordering of the objects.

[0020] Example 1 An illustrative embodiment of the present invention, such as Figure 1 As shown, Figure 1 This is a flowchart illustrating an optimization method for a bus route contracting scheme according to Embodiment 1 of the present invention. The optimization method for a bus route contracting scheme provided in this application includes: Step S100: Based on the three-way interaction logic between the system planning end, the operator end, and the passenger end in the public transport network, construct a three-layer planning model; Optionally, the three-level programming model includes: an upper-level model with the objective of minimizing the total system cost, and a decision variable that is a binary variable representing whether each bus route is contracted; a middle-level model with the objective of maximizing the net profit of the contracted route operators, and a decision variable that is the service frequency of each contracted route; and a lower-level model that is a bus network equilibrium allocation model used to simulate the passenger flow distribution at a given service frequency.

[0021] Furthermore, optionally, the optimization method for a bus route contracting scheme provided in this application embodiment further includes: constructing a lower-level model, which includes: constructing a hypergraph model based on the bus network topology, wherein the nodes of the hypergraph model include bus stops and transfer points, and the road segments include boarding road segments, in-vehicle road segments, alighting road segments, and walking road segments; introducing the concept of hyperpath to model the probabilistic choice behavior of passengers at transfer points, and based on the Wardrop equilibrium principle, describing the conditions for the network to reach an equilibrium state in the form of variational inequalities to obtain the lower-level model.

[0022] Step S102: Apply the nested generalized Benders decomposition algorithm to solve the three-level programming model to obtain the line contracting scheme and the corresponding service frequency scheme that minimizes the total system cost. Optionally, step S102, applying the nested generalized Benders decomposition algorithm to solve the three-level programming model, includes: using a logical cutting plane to process discrete decision variables and employing relaxation techniques to accelerate computation; wherein, applying the nested generalized Benders decomposition algorithm includes: combining the middle-level model and the lower-level model into an internal bilevel programming problem and solving it using the Benders decomposition method; introducing auxiliary variables and performing McCormick linearization on the bilinear terms in the objective function to transform the internal bilevel programming problem into a mixed-integer linear programming problem; combining the upper-level model and the solved internal bilevel programming problem into an external bilevel programming problem and solving it using the Benders decomposition method, and gradually approaching the global optimum by iteratively adding optimality cutting planes and feasibility cutting planes.

[0023] Further, optionally, relaxation techniques can be used to accelerate computation, including: constructing a relaxation model of the internal bi-level programming problem, transforming the internal bi-level programming problem into a system optimal allocation problem by relaxing the profit maximization constraint of the middle-level model and the user equilibrium constraint of the lower-level model; and reconstructing the feasible region of passenger flow allocation based on super-segment, expressing the super-path flow as a linear combination of segment flows.

[0024] Step S104: Based on the tested public transport network data, numerical experiments are conducted on the route contracting scheme and service frequency scheme to obtain experimental results.

[0025] Optionally, step S104 involves conducting numerical experiments on route contracting schemes and service frequency schemes based on the tested public transport network data. This includes comparing the computation time and convergence speed of the nested generalized Benders decomposition algorithm before and after applying relaxation techniques under different scale test networks and passenger flow demands; and comparing the total system cost, total passenger travel time, and total operator cost indicators of the generated optimized contracting scheme with various benchmark operation schemes.

[0026] In summary, combining steps S100 to S104, the optimization method for a bus route contracting scheme provided by this application embodiment is as follows: Step (1) Define the bus route contracting problem. Based on the strategic interaction among the three key stakeholders in the operation of the bus network—the system planner, the operator, and the passenger—a three-level planning model (TLP) is established. In step (1), the Bus Route Contracting Problem (BLCP) is defined. Based on the strategic interaction among the three key stakeholders in bus network operation—the system planner, the operator, and the passenger—a three-layer planning model is established, including the following steps: Step 1.1: Based on the public-private partnership scheme, a cooperative supply model that combines public and private services is proposed, forming the core of the bus route parameter problem; a hypergraph is established. In this graph, N represents the set of points and A represents the set of road segments, with the bus stop modeled as a transfer node. At the same time, the hypergraph contains a set of virtual nodes to represent the boarding and alighting activities of the lines passing through the station.

[0027] set up and Represent all transfer points and bus stops respectively, satisfying The route set A is divided into 4 subsets, each corresponding to a different stage of the transfer journey: the boarding route set... In-vehicle travel sections The drop-off section is a collection of and pedestrian areas ,satisfy: Each road segment a contained in road segment set A is characterized by its free-flow travel time. and road segment frequency Let L be the set of bus routes. Each route l contained in set L can be viewed as a subgraph of the network, i.e. .

[0028] Based on the hypergraph described above, the route selection behavior of passengers in the context of public transportation is described, which forms the basis for the formulaic analysis of three-level programming.

[0029] Step 1.2: Establish a three-tier decision-making structure involving three stakeholders and define their interactions. The specific steps are as follows: BLCP comprises a three-tiered decision-making structure involving three stakeholders. At the top layer, the system planner, acting as the system leader, decides which bus routes will be outsourced. A binary decision variable is defined. ,in Indicates the line Outsourced to private operators. The goal of system planning is to minimize total social cost. In the middle layer, the operator acts as an intermediate follower-leader, setting service frequencies on the outsourced lines. To maximize net profit, the outsourcing decision-making process at the system planning level is initiated. This decision considers multiple revenue and cost factors, including demand allocation, operating costs, fares, and subsidies, while being constrained by parameters such as minimum service levels and fleet size. At the lower level, passengers, as ultimate followers, choose routes that minimize overall travel costs based on public-private partnership service supply. By using public transport equilibrium assignment modeling, a state is ultimately reached where no passenger can unilaterally reduce their travel costs by changing routes, at which point the equilibrium segment flow is achieved. Node waiting time .

[0030] These three levels of decision-making are interdependent, forming a nested leader-follower game. The first leader-follower relationship is between the system planner and the operator; the former acts as the leader, shaping the institutional and parameter environment, while the latter acts as the follower, adjusting its operational strategies in response. Once the upstream decisions are determined, they collectively constitute a joint service supply, thus becoming the collective "leader" in the second-level leader-follower relationship with downstream passengers. Here, passengers adjust their route choices based on available services. This nested game theory relationship creates a tightly coupled interaction across all levels: system planner decisions shape operator behavior; operator decisions and public services shape the public transportation environment; and passenger responses, in turn, determine traffic flow distribution, system performance, and ultimately, the social costs that the system planner strives to minimize. In equilibrium, the system tends to be stable, and no stakeholder can unilaterally change decisions to improve their own outcome without disrupting system balance.

[0031] Step 1.3: Based on the travel costs of passengers, the operating costs of private operators, and the social costs of subsidies, define the total social cost function as the mathematical expression of the upper layer of the model. The specific steps are as follows: At the upper level, the system planning stage seeks to minimize the total social cost. The total social cost consists of three parts: the travel cost for passengers, the operating cost for operators, and the social cost of subsidies. The objective function is: (1) in, All are matrices. , It is a binary variable used to represent a circuit. Whether it has been contracted out; , It is a discrete variable, representing the lines operated by the carrier. Service frequency; , It is a continuous variable, representing road segments. Customer traffic; , It is a continuous variable, representing the passenger's position at the transfer node. Total waiting time.

[0032] Specifically, (2) (3) (4) (5) Equation (2) defines the passenger-side travel cost, including the total cost of travel in transit and the waiting cost. Here, This is the cost function of road segment 'a' depending on traffic flow, used to model congestion effects in public transport networks. (Coefficients) This refers to the average time value on the passenger side. Equation (3) represents the operator's operating costs, where... It is a line service frequency, It is the line length. This is the unit operating cost per vehicle kilometer. This parameter integrates various operating cost elements and can be used as a cost-effectiveness measure for operators. Equation (4) refers to the social cost of subsidy funds, and its model is set to the total subsidy amount defined in Equation (5). Proportional. The marginal cost of using public funds. Explain the tax-related welfare losses. The subsidy program is based on passenger traffic: a unit subsidy is granted for every kilometer the terminal travels to transport one passenger. The subsidy amount is adjusted based on a frequency allocation coefficient that reflects the service share of the operator.

[0033] Total social cost function The system-wide efficiency resulting from decisions made by three types of stakeholders was quantified. It is important to note that internal monetary transfers, such as ticket payments, were excluded because they have a offsetting effect from a system-wide perspective. The only upper-level decision variable is a binary parameter index. Therefore, the upper-level feasible set is defined as... .

[0034] Step 1.4: Based on the sum of ticket revenue and subsidy revenue, define a net profit function as the mathematical representation of the middle layer of the model. The specific steps are as follows: Based on the outsourcing decisions made at the system planning level, operators respond by setting service frequencies for outsourced lines, aiming to maximize their net profit. Net profit function Defined as the sum of ticket revenue and subsidy revenue minus operating costs: (6) Assuming passengers need to pay a unit fare per kilometer Ticket revenue is shared between public and private operators based on their contribution ratio according to service frequency. The operator allocates revenue to each line. above The dividends. Therefore, the total ticket revenue received by the operator. yes: (7) The decision-making process of operators is subject to the following constraints: (8) (9) (10) (11) Constraint (8) mandates that operators are only allowed to provide services on lines outsourced by the system planning end. It is a sufficiently large constant. Constraint (9) represents the constraint on the fleet size, where It is a line One-way travel time on the road This is the total available fleet size. Constraint (10) limits the line frequency to a finite set of candidate values. If the line The frequency is set as Then binary variables Equals 1, otherwise 0. Constraint (11) represents the Individual Rationality (IR) limit, that is, the operator will only operate if its expected net profit exceeds a pre-set threshold. They will only participate when necessary. To integrate into the overall three-layer model, the feasible region of the middle layer is defined as... Note that the IR constraint (11) is not included. In this case, because it depends on cross-layer interactions, it is therefore handled separately in the integration model.

[0035] Step 1.5: Facing the Public Transport Balanced Layout (BLCP) problem at the lower level of the model, the concept of hyperpath is introduced to establish a probabilistic decision-making model for passengers at transfer points, and a composite route strategy connecting origin and destination pairs is constructed. The specific steps are as follows: Given that the system planner and operator jointly provide service supply, the underlying problem is known as the Transit Equilibrium Assignment Problem (TEAP), which determines the distribution of passenger flow in the network. This application's embodiments consider congestion effects, making the perceived cost of travel dependent on the number of passengers using the same facilities. The public transportation scenario introduces specific modeling complexities, distinguishing it from standard road traffic assignment models.

[0036] The key difference lies in the common route problem: unlike drivers in road networks who typically choose a fixed path, public transport passengers waiting at stops may consider multiple overlapping routes and choose to board the first bus to arrive. To describe this behavior, the concept of hyperpath is introduced, which constructs a composite route strategy connecting origin-destination pairs (OD pairs) by modeling the probabilistic decisions of passengers at transfer points.

[0037] Based on computational requirements, via a hyperpath Defined as a complete hypergraph a subgraph ,in and These represent the node set and the road segment set of the hyperpath, respectively. Refers to the OD pair set. For each ,definition This is the set of all feasible acyclic hyperpaths connecting the origin and destination. Each hyperpath... From a pair of path probability matrices and Display features, among which and These refer to the hyperpath. Up through node and road sections The possibility. Hyperpath The collection of road sections, in order to Starting point, and assuming: Without real-time bus location information, passengers arrive randomly, and the system can accurately estimate the remaining travel time after boarding. Bus departure intervals follow an exponential distribution.

[0038] transit station Expected waiting time It can be represented as: (12) here, Directions The frequency. For a section of road where vehicles board. Its value is determined by two parties: the road segment set by the system planning end. Initial service frequency Additional service frequencies provided by the operator ,satisfy Therefore, at the transit node Used road section The probability is defined as: (13) After that, and They respectively refer to the superpath The starting point and the ending point, It refers to The collection of road sections, with Assuming the endpoint is reached, we obtain: (14) (15) (16) With these equations, the cost of a hyperpath can be defined. for: (17) In equation (17), the first term on the right represents the expected travel cost on the road segment, while the second term reflects the expected waiting cost at the transfer node. Indicates road segment The cost function is dependent on traffic flow. Since there are four types of road segments in the public transport hypergraph, representing different components of a public transport journey, their cost functions are given in equations (18)–(21). Assuming that the cost function for boarding and alighting segments is defined as dependent on traffic flow and asymmetric, this means that the cost function… It depends not only on its own traffic It also depends on the traffic flow on other related road sections. Here, This refers to the vehicle capacity related to a road segment. , , , All are non-negative parameters.

[0039] (18) (19) (20) .(twenty one) According to equation (17), the passenger's route selection behavior is modeled using Wardrop's first principles. Indicates superpath Customer traffic, Indicates OD pair The minimum travel cost, the equilibrium condition of TEAP is expressed as: (twenty two) In order to complete traffic flow assignment, OD demand must be assigned to hyperpaths. Indicates OD pair The total demand should satisfy flow conservation and non-negativity constraints. (twenty three) .(twenty four) For simplicity, let the OD demand matrix be... The hyperpath traffic matrix is and use Let represent the association matrix between hyperpaths and OD pairs. The feasible set in the hyperpath flow space is defined as: The VI problem is: (25) in It is a matrix that balances hyperpath flows. It is a matrix of superpath costs. This application embodiment requires the use of low-level decision variables. and auxiliary variables The model is restated to ensure consistency with the various model components introduced above. This application's embodiments use... Let represent the probability matrix of the superpath association of road segments. It can be represented as Use separately. This represents the correlation matrix between node waiting time and hyperpath, and its elements are defined as follows: . It can be restated as: (26) in, and These are the traffic flow on the balanced road segment and the waiting time at the node.

[0040] Step 1.6: Based on the upper, middle, and lower layer modeling framework, establish a unified three-layer planning model. The specific steps are as follows: Based on the planning of the upper, middle, and lower layers, these are integrated into a unified three-layer programming model (TLP). This problem can be formulated as: (27) Step (2) Develop a Nested Generalized Benders Decomposition algorithm based on the logical cutting plane, and further accelerate the solution through relaxation techniques; In step (2), a nested generalized Benders decomposition algorithm (N-GBD) based on the logical cut plane is developed, and the solution is further accelerated by relaxation acceleration techniques, including the following steps: Step 2.1: Introduce a series of equivalent transformations to reformulate the internal two-level problem (I-BLP), and apply Benders decomposition to solve the I-BLP problem. The specific steps are as follows: Provide a fixed upper-level decision ,but Defined as: (28) Constrained by constraints (3), (5), (6), (7), and (11), the passenger-side equilibrium condition can be expressed as a variational inequality constraint: (29) This formula transforms the original two-level model into a single-level mathematical programming (MPEC) with equilibrium constraints. While maintaining the leader-follower structure, this formula allows for direct modeling and analysis. However, from the perspective of Benders decomposition, the operator's revenue function introduces a bilinear term consisting of the product of frequency and traffic variables. These nonlinear characteristics hinder the decomposition of the problem. To address this issue, a series of equivalent transformations are introduced.

[0041] First, introduce an auxiliary variable. To indicate the private operator's side on the line Total revenue (ticket price plus subsidies): (30) Next, due to the decision variable in the denominator, the income sharing item It is non-linear. This is because it originates from discrete sets. The embodiments of this application use frequency selection variables. And restate the statement: (31) Substituting equations (30) and (31) into the net profit function, we get: (32) The remaining bilinear terms in equation (32) Linearization can be achieved using the McCormick Envelope approach. Define another auxiliary variable. To express And introduce the following constraints with sufficiently large constants. : (33) (34) (35) With these transformations, It is reformulated as a single-layer linear MPEC, eliminating all bilinear terms from the objective function: (36) Subject to constraints (29), (30), (33)-(35), and (37) Now we'll use Benders decomposition to solve this. Introduce an auxiliary continuous variable. And construct the relaxation master problem, defined as follows: : (38) here, This represents the curved cut set generated iteratively from the subproblems and added to the main problem. In each iteration, based on the candidate frequency solutions generated from the main problem... Solve related sub-problems It determines passenger flow and auxiliary variables: (39) Subject to constraint (30) and the following constraints: (40) (41) Although constraint (40) introduces a variational inequality that transforms the subproblem into MPEC, its special structure requires an efficient solution.

[0042] Since the subproblems are not linear, embodiments of this application employ an L-shaped method based on binary properties. Generate an effective cutting plan. Define For subproblems The optimal value. By introducing a set. and complementary sets The following optimal cuts need to be included in the main problem: (42) in yes about The upper boundary.

[0043] On the other hand, if the subproblem is infeasible, which means that the current frequency solution violates the IR constraint (37), the following feasible cut should be added to exclude the solution from the feasible solution set of the main problem: (43) The complete iterative process is summarized in Algorithm 1.

[0044] Algorithm 1 Step 2.2: Define the external two-layer problem (O-BLP) and apply Benders decomposition to solve the O-BLP problem. The specific steps are as follows: Solving the outer-layer bi-level problem (O-BLP) is similar to solving the inner-layer problem (I-BLP), involving an iterative process of alternating between the main problem and nested subproblems. This process is refined by progressively adding optimality and feasibility-based cutting conditions. The key difference is that I-BLP is now used as a subroutine within the O-BLP solution algorithm to evaluate the subproblems. For simplicity, repeated technical details will be omitted below.

[0045] BLP is defined as: (44) Subject to constraints (2)-(5), (11), and: (45) (46) Constraint (45) ensures that the net profit of the private operator is within the range given by the outsourcing decision in the system planning stage. The following is maximized, while constraint (46) ensures passenger-side equilibrium. To apply Benders decomposition, this embodiment introduces an auxiliary variable. And define the relaxation master problem. for: (47) in This represents the set of Benders cuts accumulated during the iteration process. For a given solution to the main problem, The corresponding sub-problems are: (48) Subject to constraints (2)-(5), (11), and (49) (50) Due to decision variables It is binary; this application's embodiments again apply integer-based Benders cutting. Definition For subproblems The optimal value, and They represent Outsourced and non-outsourced lines, and in Above all Give a lower bound If the subproblem returns an optimal solution, then add the following optimality cut in a manner similar to constraint (42): (51) If a subproblem is infeasible, it means that the recent system planning outsourcing plan cannot provide the operator with a sufficient profit margin because no frequency setting scheme can reach the minimum profitability threshold. In this case, feasibility segmentation needs to be added: (52) To further improve computational efficiency, symmetry-breaking constraints are introduced. Definition For set The minimum value. In each iteration, when a new solution is obtained. When this happens, the following cuts will be added. In the question: (53) Cut (53) ensures that if a line is outsourced by the system planning end, the operator must allocate a strictly non-negative frequency to it. The complete algorithm is listed in Algorithm 2.

[0046] Algorithm 2 Step 2.3: An acceleration strategy based on sub-problem evaluation is proposed. This involves constructing relaxed sub-problems, reformulating the feasible region based on the start and end points, and separating the objective function. This significantly reduces the number of two-level sub-problems that must be solved precisely, thus accelerating the algorithm's execution. The specific steps are as follows: Although the proposed nested GBD algorithm theoretically guarantees convergence to the global optimum within a finite number of iterations, preliminary experiments show that its actual running time is too long. To alleviate this problem, this application proposes an acceleration strategy based on subproblem approximation evaluation.

[0047] To relax The model in this application eliminates the hierarchical structure and instead treats all decisions as being coordinated within a system-optimal framework. Specifically, this includes: (1) relaxing constraints (49) to remove the profit maximization objective of the private operator; (2) relaxing constraints (50) to allow passenger flow to follow the system-optimal (SO) principle, rather than user equilibrium (UE); and (3) relaxing IR constraints (11). These relaxations lead to the following problems, defined as : (54) In addition to these simplifications, It remains computationally challenging. To avoid enumerating all hyperpaths, embodiments of this application propose a feasible set based on OD traffic on generalized network elements called hypersegments. Alternative expression. A hyperpath is defined as a tuple. ,in Indicates the starting point, while It is the endpoint set. Unlike regular road segments, supersegments allow transitions from one node to multiple potential downstream nodes. Each supersegment... With a set of underlying regular road segments Related. To describe the probabilistic nature of passenger-side decisions in overtaking segments, embodiments of this application allocate a set of path probabilities. To each overpass section This indicates the section when crossing the overpass. The probability of being selected. These probabilities satisfy: (1) as well as (2) make Indicates the relationship with the node Crossing over section The OD (Original Demand) pairs are related to passenger flow. Using this hyperpath-based flow representation, the feasible set... It can be restated as: (55) (56) (57) (58) (59) Constraint (55) is a flow conservation constraint, which relates to OD-based hyperpath flow, where Represents the set of all out-of-way segments, Starting from this point, constraint (56) concretizes the path probability as a function related to the line frequency. Constraints (57) and (58) establish the over-segment flow and the segment flow, respectively. and node waiting time The relationship between the two variables. Constraint (59) defines the nonnegative decision variables.

[0048] Now we deal with the remaining bilinear terms, which involve the frequency variable. Traffic flow variables of road segments The product of. In principle, because It takes values ​​from a finite discrete set, and all possible realizations can be listed. The proposed solution involves solving the corresponding subproblems separately. However, this method is computationally infeasible.

[0049] To facilitate this decomposition and utilize monotonicity, embodiments of this application employ a method that decomposes from feasible sets... Abandoning fleet size constraints (9) and further relaxing them This is to ensure that the monotonicity argument applies without cross-line correlation. The relaxed feasible set is defined as... ,make , and Let represent the optimal solutions to the three subproblems under the relaxed setting. Their sum yields an efficient and computationally effective lower bound for the optimal value. A detailed explanation of these three subproblems follows: Sub-problem 1: Minimize travel costs.

[0050] (60) Corresponding to the first subproblem, this problem aims to minimize passenger-side travel costs and is essentially a traffic assignment model following the SO principle. The key characteristic of this subproblem lies in its objective function. Relative to variables It exhibits a monotonically decreasing relationship. Therefore, when When taking the maximum feasible value, Minimum. Defined as a set The maximum value in, if ,make ,otherwise Let the matrix .fixed , Reduced to the following simplified problem, defined as Only of them Still decision variables: (61) Such problems can be effectively solved using the classic outer envelope (OA) method. In this embodiment, the nonlinear objective function is linearized at selected points using a tangent hyperplane, iteratively constructing a mixed-integer linear programming (MILP) subproblem. Due to the convexity of the nonlinear term, under the condition of mild regularity, the OA process can generate a series of increasingly tighter lower bounds and converge to the global optimum. Modern solvers such as Gurobi have built-in support for OA technology; therefore, this embodiment relies on the solver's implementation scheme and will not elaborate on the algorithm details.

[0051] Sub-problem 2: Minimize operating costs.

[0052] (62) This corresponds to the second subproblem. Objective function. Depends only on the frequency variable ,about Monotonically increasing. Therefore, when Taking the minimum value yields the minimum function value. Let... Define a set The minimum value. Let Define the line The minimum allowable frequency, and define the matrix. If ,but ,otherwise Since the optimal value can be obtained by direct substitution, this subproblem does not require explicit optimization.

[0053] Sub-problem 3: Minimizing social capital costs.

[0054] (63) This corresponds to the third sub-problem. Objective function. It depends on frequency and flow rate variables. exist The condition is monotonically increasing on the feasible set, indicating that when... When taking the minimum feasible value, we get The minimum value. Therefore, fixed. , The problem is reduced to the following simplified problem, which is clearly a MILP problem and can be solved directly using a solver: (64) Step (3) Conduct numerical experiments to verify the effectiveness of the algorithm.

[0055] In step (3), numerical experiments are conducted to verify the effectiveness of the algorithm, including the following steps: Step 3.1: Using Mandl's network as the experimental benchmark, change two dimensions to generate 20 test cases and evaluate the efficiency of the Nested-GBD algorithm and its acceleration method; Step 3.2: Compare the performance of various operating schemes and evaluate the effectiveness of selective public-private partnership models for public transportation networks; Step 3.3: Conduct a comprehensive sensitivity analysis on several key parameters that affect the behavior of the three levels of stakeholders and the equilibrium state of the system, and explore how changes in these parameters affect service supply, passenger welfare, and the efficiency of the overall system.

[0056] like Figure 2 As shown, Figure 2 This is a schematic diagram of the technical route in an optimization method for a bus route contracting scheme according to Embodiment 1 of the present invention. To design route contracting parameters that minimize social costs, this application provides a three-level programming model. Furthermore, to accurately solve this model, this application provides a nested generalized Benders decomposition algorithm based on a logical cutting plane. This application verifies the proposed three-level programming model and its solution algorithm from multiple perspectives through numerical experiments, examining the computational performance of the nested GBD algorithm and its acceleration strategy; evaluating the effectiveness of the proposed selective PPP model by comparing alternative operating schemes; and employing sensitivity analysis to explore how changes in key parameters affect network equilibrium and stakeholder decisions, with a focus on providing opinions for management decisions. This application uses a Mandl network as an experimental benchmark example, such as... Figure 3 As shown, Figure 3 This is a baseline example diagram of a numerical experiment in an optimization method for a bus route contracting scheme according to Embodiment 1 of the present invention. The network has 15 nodes and 42 directed segments. Each segment is labeled with its free-flow travel time (in minutes), and the node index is also labeled accordingly. It is assumed that the system planner currently operates 5 bidirectional bus routes, and their details, including stop order, one-way travel time, and initial service frequency, are listed in Table 1. In this setup, if a route is outsourced by the system planner, the operator may operate on any one of the routes. The candidate frequency set is defined as... (Number of vehicles per hour). The original dataset contains a highly dense demand matrix. To strike a balance between problem size and computational feasibility, this embodiment retains the 50 OD pairs with the highest travel demand. The parameters used, their default values, and references are listed in Table 2.

[0057] Table 1. Detailed information on bus routes Table 2 Default parameters set in numerical experiments All experiments were performed using Python 3.8 on a personal computer configured with an Intel i9-12900H CPU and 32 GB of RAM. During the solution process, the commercial solver Gurobi 10.1 was used whenever necessary to solve mixed integer programming (MILP) and linear programming (LP) problems. The specific steps are as follows: Step 3.1: Using Mandl's network as the experimental baseline, two dimensions were changed to generate 20 test instances to evaluate the efficiency of the Nested-GBD algorithm and its acceleration method. Each instance was labeled as... ,in Indicates the number of lines. This is the scale factor applied to the requirements. Table 3 shows the calculation results. The Obj column lists the optimal target value for each test instance. The Nested-GBD and Nested-GBD-accel columns list the solution time of the Nested-GBD algorithm and the upgraded version with the applied acceleration strategy, respectively. The Accel.Rate column shows the acceleration ratio.

[0058] Table 3. Calculation results with and without acceleration technology The results show that the solution time increases with the number of routes and the level of demand, which is expected. The dimensionality of the decision variables increases, and solving nested traffic assignment problems based on user entities incurs computational burden. The proposed acceleration strategy brings significant improvements in most instances, especially in high-complexity scenarios. For example, in L5-1.8 and L5-2.0, the acceleration strategy reduces computation time by more than 40%. These gains mainly stem from the subproblem relaxation mechanism—by establishing a compact lower bound, the algorithm skips some high-cost I-BLP evaluations. This mechanism avoids redundant two-level solution processes while maintaining convergence guarantees, and the effectiveness of the relaxation-enhanced decomposition strategy has been fully verified in practice.

[0059] Step 3.2: Compare the performance of various operating schemes and evaluate the effectiveness of the selective public-private partnership model for public transport networks.

[0060] The alternative operating solutions include: (1) Selective PPP: The solution proposed in the embodiments of this application.

[0061] (2) Open-access PPP: In the comparison scheme, the system planning end opens all lines to the operator for contracting. This corresponds to setting up... It should be noted that in this series of experiments, to ensure that the obtained solution represents the "true" optimal frequency setting of the operator, the valid inequality (53) has been removed.

[0062] (3) Public Operation: A comparative scheme in which there is no route outsourcing, and the entire public transport service is operated by the system planning end. This corresponds to setting up .

[0063] (4) Private Operation: A comparative approach where the system planning end completely withdraws from public transport operations, and all routes are contracted and operated by the operator. The operator can independently determine their service frequency, which corresponds to setting... .

[0064] Table 4 shows the optimal decision and outcome metrics for each scheme. The columns include data from the upper-level system planning stage (…). The optimal route outsourcing result comes from the mid-level operator ( The frequency settings and a series of outcome metrics: total social cost (Obj.), passenger-side travel cost ( ), operator operating costs ( Social capital costs ( ), and net profit of operators ( ).

[0065] Table 4 Comparison of Four Operating Plans The results in Table 4 first reveal that the public operation scheme, reflecting the initial state of the system, resulted in the highest total social cost ($9496.21), entirely attributable to passenger travel costs. This confirms the inefficiency of low-frequency public operation services and highlights the performance limitations of purely system-planned operation systems. Introducing operator involvement significantly improved network efficiency. Under both public-private partnership and full privatization models, the service frequency of high-demand routes was increased, thereby reducing travel time.

[0066] However, complete privatization is not an ideal solution either. Although its total social cost is significantly lower than the public operation model ($6,762.91 vs. $9,496.21), its performance is still inferior to the two public-private partnership models. The root cause lies in the profit-maximizing behavior of the operators: they concentrate their fleets on a few high-demand, heavily subsidized, and profitable routes, while neglecting routes with lower profitability but social value. For example, in the scenario of complete privatization, the complete shutdown of Route A leads to severe service gaps in parts of the network. Although the operators achieve record net profits, the passenger travel costs are as high as $5,454.56, exposing the misalignment between private incentives and public welfare.

[0067] The two PPP-based schemes can more effectively balance these conflicting objectives. In particular, the proposed selective PPP scheme achieves the best overall performance, with a total social cost of $4,329.52, 14% lower than the open PPP scheme ($5,055.53). This cost reduction is primarily due to a decrease in passenger travel costs ($3,495.32 vs. $4,180.67), indicating that selective regulation of route accessibility helps create a fairer and more efficient service model. A deeper analysis of departure frequency settings reveals that under the selective PPP model, operators, guided by the system, deploy their fleets on routes A, B, and C, with route B receiving the highest departure frequency. In contrast, under the open PPP scheme, although operators are aware of the significant social benefits of route B, they choose to completely avoid it, instead concentrating their frequency on route E, which may offer higher returns in a competitive environment. This result exhibits an effect similar to the Braess paradox: increased decision-making freedom leads to worse systemic outcomes due to behavioral dissonance. In the open PPP scheme, operators must anticipate the residual effects of system planning in their deployment optimization. This strategic consideration might lead it to abandon certain routes where public presence reduces marginal profits, even if these routes are crucial from a system-wide perspective. In contrast, by narrowing the range of parameterized routes, selective PPP schemes allow system planners to indirectly guide operator behavior in a socially desirable direction without directly controlling operational variables.

[0068] Step 3.3: Conduct a comprehensive sensitivity analysis on several key parameters affecting the behavior of the three levels of stakeholders and the system equilibrium state, and explore how parameter changes affect service supply, passenger welfare, and the efficiency of the overall system. The parameters to be tested include: (1) subsidy coefficient (2) Ticket price and (3) Cost factors on the operator's side .

[0069] (1) Subsidy coefficient Subsidies are a primary means of incentivizing private sector participation in public transport service provision. To test the effect, the parameter was adjusted from 0 to 4.0 in increments of 0.1, generating 41 test cases for each of the L4-1.0 and L5-1.0 instances. Figure 4 This is a schematic diagram illustrating the impact of the subsidy coefficient on the outcome indicators in an optimization method for a bus route contracting scheme according to Embodiment 1 of the present invention. Figure 4 As shown, Figure 4 The results indicators under the system equilibrium state are shown as follows The changing pattern. With... Increased availability has led to stronger financial incentives for operators, resulting in higher service frequency across outsourced lines. Improved service availability reduces travel time and enhances network performance, reflected in lower passenger travel costs (Pax.). However, these improvements come at the cost of increased operator operating costs (Oper.), and rising subsidies for system planning also contribute to the overall social cost.

[0070] Overall social costs do not exhibit monotonicity. Firstly, they decrease... At this point, a minimum is reached. This inflection point represents the optimal subsidy level; beyond this level, funding becomes ineffective. When subsidies are low, operators lack sufficient incentive to provide services limited to basic needs, typically covering only high-demand lines. As subsidies increase, operators respond more actively by expanding service frequencies and delivering significant benefits to passengers that far outweigh the financial burden. However, once operators saturate their response by allocating the highest frequencies to the most profitable lines, any additional subsidies only increase public spending without providing additional travel time savings. Furthermore, further increases in subsidies do not alter the high- and mid-level solutions, i.e., they cannot drive the network towards complete outsourcing, confirming the conclusion that selective contracts help curb inefficiencies stemming from profit-driven behavior.

[0071] (2) Ticket price The fare increased from 0 to 3.0 in increments of 0.1, generating 31 test cases. These tests were conducted on the L4-1.0 and L5-1.0 examples. In the current setting, the fare and subsidies jointly determine the revenue streams for private operators. Therefore, the strategic behavior of operators in response to fare changes is similar to that observed under subsidy changes.

[0072] Figure 5 This is a schematic diagram illustrating the impact of fares on outcome indicators in an optimization method for a bus route contracting scheme according to Embodiment 1 of the present invention. Figure 5 As shown, Figure 5The results show that as fares rise, operators are more willing to increase service frequency to attract more passengers. This leads to lower travel costs for passengers, but higher operating costs and social capital costs for operators. Notably, unlike the subsidy scenario, total social costs decrease monotonically with rising fares, eventually leveling off. This difference stems from the fact that fares are essentially an internal revenue transfer between passengers and private operators, rather than a net expenditure from a social cost perspective. Once the system reaches equilibrium, further fare increases will only transfer surplus value from passengers to operators, without affecting efficiency indicators such as departure frequency or coverage.

[0073] These findings also provide valuable guidance for fare design in system planning. At relatively low fare levels, moderate price adjustments can both increase operator participation and improve overall system performance. Passengers are willing to pay slightly higher fees for significantly better service. However, beyond a certain threshold, price increases will no longer improve system operation; they will only cause allocation distortions and may impair accessibility and fairness. Therefore, pricing mechanisms should not only be viewed as financial tools but also as a means to balance the efficiency and fairness of public transportation services.

[0074] (3) Cost factors of operators Cost factors This determines the operating costs that the operator must bear to provide each service. To examine its impact on system balance, the cost is increased by an increment of 0.1. From version 0 to 4.0, 41 test cases were created. Experiments were conducted on L4-1.0 and L5-1.0 instances.

[0075] As expected, Figure 6 This is a schematic diagram illustrating the impact of operator-side cost factors on outcome indicators in an optimization method for a bus route contracting scheme according to Embodiment 1 of the present invention. Figure 6 As shown, Figure 6 The results show a strong monotonic relationship. Lower cost coefficients allow operators to provide more frequent services while maintaining a given level of profitability, thereby expanding coverage and improving passenger benefits. Conversely, as cost coefficients rise, operators will reduce service frequency or exit less profitable routes, leading to decreased overall accessibility and increased travel time for users.

[0076] From a systems planning perspective, total social costs decrease continuously as operating costs decline, further confirming the necessity of introducing efficient private partners. It's worth noting that while subsidies and pricing mechanisms can mitigate the negative impacts of high costs, their effectiveness has limits. If operating costs are too high, even substantial subsidies may fail to enhance the attractiveness of partnerships, ultimately leading to service gaps and network performance degradation. These findings suggest that operational efficiency should be a core consideration when selecting or regulating private operators. Partnering with low-cost operators can improve service effectiveness and expand the scope of systems planning solutions, allowing for flexible adjustments to fares and subsidies to achieve desired network goals.

[0077] This application provides an optimization method for bus route contracting schemes. Outsourcing bus services to operators has become an important strategy for improving the efficiency and reducing costs of urban public transportation systems. However, existing research generally overlooks the differences at the route level and the potential benefits of selectively outsourcing only a portion of the bus network. This application's optimization method for bus route contracting schemes aims to minimize total social costs by strategically determining outsourced bus routes at the system planning stage, while considering the behavioral responses of both operators and passengers. The decision problem is modeled as a three-level programming model, integrating three strategic interactions: route contracting decisions at the system planning stage, departure frequency settings by profit-maximizing private operators, and route selection by passengers based on a public transportation equilibrium allocation model. To accurately solve the model, a nested generalized Benders decomposition algorithm based on a logical cutting plane is developed and further optimized using relaxation acceleration techniques. Computational experiments verify the effectiveness of the algorithm and reveal important management implications: selective bus route contracting is superior to purely public and purely private operation models, and the system equilibrium state is highly sensitive to exogenous parameters. These findings emphasize the importance of implementing coordinated system-level supervision in public-private partnership design for public transportation planning.

[0078] This invention employs the above technical solution, constructing a three-layer planning model based on the three-way interaction logic between the system planner, operator, and passenger in a public transport network. It then applies a nested generalized Benders decomposition algorithm to solve the three-layer planning model, obtaining the route contracting scheme and corresponding service frequency scheme that minimizes the total system cost. Based on tested public transport network data, numerical experiments are conducted on the route contracting scheme and service frequency scheme. The experimental results, compared with existing technologies, demonstrate the following technical advantages: It accurately describes the strategic interactions between the system planner, operator, and passenger, and efficiently solves the problem using a nested decomposition algorithm, ultimately outputting the optimal route contracting and frequency setting scheme that minimizes the total social cost.

[0079] Example 2 An illustrative embodiment of the present invention, such asFigure 7 As shown, Figure 7 This is a schematic diagram of an optimization device for a bus route contracting scheme according to Embodiment 2 of the present invention. The optimization device for a bus route contracting scheme provided in this application includes: The construction module 72 is used to construct a three-layer planning model based on the three-way interaction logic between the system planner, operator, and passenger in the public transport network; the calculation module 74 is used to solve the three-layer planning model by applying the nested generalized Benders decomposition algorithm to obtain the route contracting scheme and the corresponding service frequency scheme that minimizes the total system cost; the verification module 76 is used to conduct numerical experiments on the route contracting scheme and service frequency scheme based on the test public transport network data to obtain the experimental results.

[0080] This invention employs the above technical solution, constructing a three-layer planning model based on the three-way interaction logic between the system planner, operator, and passenger in a public transport network. It then applies a nested generalized Benders decomposition algorithm to solve the three-layer planning model, obtaining the route contracting scheme and corresponding service frequency scheme that minimizes the total system cost. Based on tested public transport network data, numerical experiments are conducted on the route contracting scheme and service frequency scheme. The experimental results, compared with existing technologies, demonstrate the following technical advantages: It accurately describes the strategic interactions between the system planner, operator, and passenger, and efficiently solves the problem using a nested decomposition algorithm, ultimately outputting the optimal route contracting and frequency setting scheme that minimizes the total social cost.

[0081] Example 3 The present invention provides a storage medium storing a computer program, wherein the computer program is configured to execute the above-mentioned optimization method for bus route contracting scheme when running.

[0082] The above description is merely a preferred embodiment of the present invention and does not limit the implementation and protection scope of the present invention. Those skilled in the art should realize that any equivalent substitutions and obvious changes made based on the description and illustrations of the present invention should be included within the protection scope of the present invention.

Claims

1. An optimization method for a bus route contracting scheme, characterized in that, include: Based on the three-way interaction logic between the system planner, operator, and passenger in the public transport network, a three-layer planning model is constructed. The nested generalized Benders decomposition algorithm is applied to solve the three-level planning model to obtain the line contracting scheme and the corresponding service frequency scheme that minimizes the total system cost. Based on the tested public transport network data, numerical experiments were conducted on the route contracting scheme and the service frequency scheme, and the experimental results were obtained.

2. The optimization method for the bus route contracting scheme according to claim 1, characterized in that, The three-level programming model includes: The upper-level model aims to minimize the total system cost, and the decision variables are binary variables representing whether each bus route is contracted out. The intermediate-level model aims to maximize the net profit of the operators who contract the lines, with the service frequency of each contracted line as the decision variable. The lower-level model is a public transport network equilibrium allocation model, used to simulate passenger flow distribution at a given service frequency.

3. The method for optimizing the bus route contracting scheme according to claim 2, characterized in that, The method further includes: Constructing the lower-level model includes: A hypergraph model is constructed based on the topology of the public transport network. The nodes of the hypergraph model include bus stops and transfer points, and the road segments include boarding road segments, in-vehicle road segments, alighting road segments, and walking road segments. The concept of hyperpath is introduced to model the probabilistic choice behavior of passengers at transfer points, and based on the Wardrop equilibrium principle, the conditions for the network to reach equilibrium are described in the form of variational inequalities, thus obtaining the lower-level model.

4. The method for optimizing the bus route contracting scheme according to claim 2, characterized in that, The application of the nested generalized Benders decomposition algorithm to solve the three-level programming model includes: Discrete decision variables are processed using a logical cutting plane, and relaxation techniques are employed to accelerate computation. The application of the nested generalized Benders decomposition algorithm includes: The middle-level model and the lower-level model are combined into an internal bi-level programming problem, which is solved using the Benders decomposition method. Auxiliary variables are introduced and the bilinear terms in the objective function are linearized using McCormick, thus transforming the internal bi-level programming problem into a mixed-integer linear programming problem. The upper-level model is combined with the solved internal bi-level programming problem to form an external bi-level programming problem, which is solved using the Benders decomposition method. The global optimal solution is gradually approached by iteratively adding optimality cut planes and feasibility cut planes.

5. The method for optimizing the bus route contracting scheme according to claim 4, characterized in that, The use of relaxation techniques to accelerate computation includes: A relaxed model of the internal bi-level programming problem is constructed. By relaxing the profit maximization constraint of the middle-level model and the user equilibrium constraint of the lower-level model, the internal bi-level programming problem is transformed into a system optimal allocation problem. Based on the reconstructed feasible region of passenger flow allocation from super-segment, super-path flow is expressed as a linear combination of segment flow.

6. The method for optimizing the bus route contracting scheme according to claim 1, characterized in that, The numerical experiments conducted on the route contracting scheme and the service frequency scheme based on the test-based public transport network data include: Compare the computation time and convergence speed of the nested generalized Benders decomposition algorithm before and after applying relaxation techniques under test networks of different sizes and passenger flow requirements. The system's total cost, total travel time for passengers, and total cost for operators are compared with the generated optimized contracting scheme and various benchmark operating schemes.

7. An optimization device for a bus route contracting scheme, characterized in that, include: The module is used to build a three-layer planning model based on the interaction logic between the system planner, the operator, and the passenger in the public transportation network. The calculation module is used to solve the three-level planning model by applying the nested generalized Benders decomposition algorithm to obtain the line contracting scheme and the corresponding service frequency scheme that minimizes the total system cost. The verification module is used to conduct numerical experiments on the route contracting scheme and the service frequency scheme based on the tested public transportation network data, and obtain the experimental results.

8. A storage medium, characterized in that, The storage medium stores a computer program, wherein the computer program is configured to execute the optimization method of the bus route contracting scheme according to any one of claims 1 to 6 when it runs.

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