A method for optimizing the recovery sequence of a damaged road network based on user equilibrium flow distribution with the maximum resilience as the target

Abstract

The application discloses a kind of based on user balance flow distribution with maximum toughness as target damaged road network recovery order optimization method, to solve the problem of low efficiency, decision lacks scientific basis after disaster traffic network recovery.The method comprises the following implementation steps: first, construct damaged road network toughness evaluation framework, in combination with traffic network topological structure and flow demand, define toughness index as the reciprocal of recovery time and total travel time product;Second, establish damaged road network toughness optimization scene parameter, abstract road network into connected graph and define recovery scheme set;Then, construct the mixed integer linear programming model with maximum toughness as target, convert nonlinear problem into linear problem by cutting line approximation, and set flow balance, travel demand and other constraint conditions;Finally, the model is globally optimized and solved using high-performance solver, to generate the best recovery scheme.The method of the application can scientifically quantify road network toughness and optimize recovery order, avoid the blindness of experience decision, quickly improve the traffic efficiency of post-disaster traffic system, provide efficient and scientific decision support for post-disaster traffic recovery, and have important practical application value.

Description

Technical Field

[0001] This invention provides a MILP method for optimizing the restoration order of damaged road networks based on user-balanced traffic allocation, with the goal of maximizing resilience. This invention relates to the field of traffic network restoration technology, specifically a method for optimizing the restoration order of damaged road networks based on user-balanced traffic allocation, with the goal of maximizing resilience. This method is applicable to rapid restoration planning of traffic networks after disasters. Through mathematical modeling and optimization algorithms, it determines the optimal restoration sequence of the damaged road network to improve the overall resilience and traffic efficiency of the network. This invention combines traffic engineering, operations research, and computer science, using a mixed-integer linear programming model to quantitatively analyze the road network restoration problem. It is applicable to fields such as urban traffic management, emergency response, and infrastructure repair. Its core technology lies in evaluating road network resilience through a user-balanced traffic allocation model and optimizing the restoration order, thereby maximizing the network's capacity in the shortest time and providing scientific decision support for post-disaster traffic recovery. Background Technology

[0002] Following natural disasters or man-made catastrophes, transportation networks often suffer severe damage, leading to reduced traffic capacity and hindering rescue efforts and normal urban operations. Traditional road network restoration methods largely rely on experience or simple priority rules, lacking systematic optimization and struggling to achieve efficient restoration in complex scenarios. Existing technologies, particularly resilience assessment and restoration sequence optimization, are primarily based on static analysis, failing to adequately consider dynamic changes in traffic flow and user behavior, resulting in restoration plans that are out of sync with actual needs. Furthermore, nonlinear road resistance functions and flow balance constraints increase the complexity of model solutions, making it difficult for existing methods to balance computational efficiency and global optimality. While recent research has attempted to combine user-balanced flow distribution models and optimization algorithms, the following problems remain: first, the definition of resilience is unclear, failing to quantify the impact of restoration sequence on road network efficiency; second, model nonlinearity makes solution difficult, hindering application to large-scale road networks; and third, there is a lack of comprehensive optimization of restoration time and traffic efficiency. Therefore, a scientific and efficient method for optimizing road network restoration sequence is urgently needed to improve the rapid recovery capability of post-disaster transportation systems. To address the aforementioned problems, this invention proposes an optimization method based on mixed-integer linear programming, providing a new technical approach for the restoration of damaged road networks. Summary of the Invention

[0003] (1) The purpose of this invention:

[0004] The purpose of this invention is to provide an optimization method for the restoration order of damaged road networks based on user-balanced traffic distribution. By quantifying road network resilience and optimizing the restoration sequence, it addresses the problems of low efficiency and poor adaptability in traditional methods. First, this invention defines a scientific resilience index based on the traffic network topology and traffic demand, quantifying the impact of the restoration order on road network traffic efficiency and providing a basis for decision-making. Next, this invention establishes an efficient optimization model, transforming the nonlinear road resistance function and traffic balance constraints into a mixed-integer linear programming problem, reducing solution complexity and ensuring the model's applicability in large-scale road networks. Finally, relying on a high-performance solver, this invention can efficiently obtain the optimal restoration plan, ensuring that the road sections that most significantly improve road network capacity are repaired within a limited time after a disaster, minimizing traffic interruption time and achieving rapid restoration decisions. Through these objectives, this invention aims to provide a scientific and efficient technical means for post-disaster traffic recovery, improving the overall resilience and emergency response capability of road networks.

[0005] (2) Technical solution: Based on the above theories and ideas, this invention provides an optimization method for the restoration order of damaged road networks with the goal of maximizing resilience, based on user balanced flow distribution, namely, a MILP-based method, the specific implementation steps of which are as follows:

[0006] Step 1: Construction of a framework for assessing the resilience of damaged road networks;

[0007] Specifically, based on a comprehensive assessment of the damage to the traffic network topology and traffic demand after a disaster, and using a user-balanced flow distribution model, a method for assessing the resilience of the damaged road network is determined. First, a common road resistance function is given, which represents the relationship between the travel time t of a road and the traffic flow f on that road segment. A common road resistance function is shown below, and this function is used as an example in this invention:

[0008] t=t0[1+α(f / c) β (1)

[0009] The objective function for balancing user traffic allocation is given below, which is the integral of travel time over traffic volume:

[0010]

[0011] Assume there are k damaged road sections, and the recovery time required for each damaged road section is τ. k For each damaged road segment restored, the total travel time for all traffic demands across the entire road network should decrease. Therefore, the reciprocal of the sum of the total travel time obtained from restoring each damaged road segment and the required restoration time can be used as the resilience R of the entire road network.

[0012]

[0013] Step 2: Define the parameter representation for the damaged road network resilience optimization scenario;

[0014] Specifically, the transportation network is mathematically abstracted as a connected graph G = (V, E), where i, j ∈ V, representing the index of a road node, i, j = 1, 2, ..., |V| (|*| represents the total number of elements in the set). Nodes correspond to intersections and key locations in the transportation network with traffic demand, such as schools, factories, hospitals, banks, office buildings, and residential areas. The set E' represents the set of damaged roads. Set K represents the set of repair sequence numbers for repairing damaged road sections, k∈K, k=1,2,…,|K|, and |K|=|E'|. Set P represents the set of repair plans, p∈P, where each element corresponds to an arrangement of the repair order for damaged road sections. Parameter h ijpk The value is either 0 or 1, indicating whether edge (i,j) has been restored when the restoration scheme p reaches the k-th edge. If it has been restored, h... ijpk =1, otherwise h ijpk =0; correspondingly, τ pk This represents the recovery time required to recover the k-th edge for recovery scheme p. Each edge (i,j)∈E in the graph represents a road in the traffic network, t 0ij c represents the free-flow time of this edge, that is, the shortest time to traverse this edge; ij Let t represent the capacity of edge (i,j), which is the maximum number of vehicles that can pass through this road within a certain period (usually one day). For the damaged road section, t 0ij and c ij These represent the free-flow time and capacity of the road section after repair. α is a parameter in the road resistance function, typically taking a value of 0.15; β is a parameter in the road resistance function, typically taking a value of 4; γ ij The parameters introduced when performing a secant approximation on the impedance function are α, β, and c. ij The parameters were calculated together. and These represent the slope and intercept of the secant line introduced when approximating the impedance function using the secant line, respectively. Let d represent the slope of the secant line when approximating edge (i,j); Let represent the intercept of the d-th secant when approximating edge (i,j). Let W represent the set of travel demands, w∈W, w=1,2,…,|W|. (s w ,t w ) represent the origin and destination of the w-th travel request, respectively. w ,t w ∈V. Parameter q wLet M represent the flow rate of the w-th trip demand. M is a large number.

[0015] Step 3: Establish a mixed-integer linear programming model for the restoration order of the damaged road network with the goal of maximizing resilience. The decision variables are the traffic flow on each road segment, the resilience corresponding to each restoration scheme, and the integral time obtained at each restoration step. The optimization objective is to maximize the restoration resilience (processed as minimum in the mathematical expression).

[0016] Specifically, in this problem, the total travel time will decrease for each damaged road segment restored. Our goal is to restore the road segments that will maximize the improvement in road network efficiency in the shortest possible time. Since we define road network efficiency as total travel time, the goal is to restore the road segments that will reduce the total travel time the most after restoration. Therefore, the objective function is defined as restoration time × total travel time. The goal is to minimize this objective function. Correspondingly, resilience is defined as the reciprocal of this objective function.

[0017] Step 3.1: Define the decision variables for optimizing the restoration order of the damaged road network with the objective of maximizing resilience;

[0018] The non-negative continuous variable represents the flow rate of the w-th pair of travel demands on road segment (i,j) when the k-th edge is restored, for the restoration scheme p.

[0019] f ijpk : A non-negative continuous variable, representing the total traffic flow on road segment (i,j) when the path is restored to the k-th edge for restoration scheme p;

[0020] t ijpk : A non-negative continuous variable, representing the total traffic flow on road segment (i,j) when f ijpk The passage time on this section of road;

[0021] y ijpk : Non-negative continuous variables, intermediate variables introduced when linearizing nonlinear impedance functions;

[0022] g ijpk : Non-negative continuous variables, intermediate variables introduced when linearizing nonlinear impedance functions;

[0023] T pk : A non-negative continuous variable, representing the total travel time of all traffic demands on the road network when the recovery scheme is restored to the k-th edge for the recovery scheme p;

[0024] B p : A non-negative continuous variable, representing the reciprocal of the recovery resilience for recovery scheme p;

[0025] Step 3.2: Establish an objective function for optimizing the restoration order of the damaged road network with the goal of maximizing resilience;

[0026]

[0027] Among them, B p T is the reciprocal of the resilience of the recovery scheme p. pk τ represents the total travel time of all traffic demands on the road network when restoring to the k-th edge for recovery scheme p, reflecting the traffic efficiency of the road network. pk This means that for recovery scheme p, the recovery time required to recover the k-th edge is minimized by taking the product of these edges, which reflects the goal of rapidly improving the traffic efficiency of the road network.

[0028] Step 4: Constrain the model by taking into account factors such as traffic demand, traffic balance, and travel impedance; Steps 2 and 3 determined the model's parameters, variables, and optimization objectives, respectively, and this step imposes multi-dimensional constraints on the model.

[0029] Step 4.1: Establish travel demand constraints;

[0030]

[0031]

[0032] in, For the wth pair of travel needs, starting from origin s w Outflow of traffic For the w-th pair of travel demand inflow starting point s w The flow rate, the first expression in constraint (5) represents the flow rate from the starting point s. w The sum of outflows should equal the flow q required for the w-th trip demand. w The second formula represents the inflow starting point s. w The total flow should be 0. Similarly, the first expression in constraint (6) represents the inflow to the endpoint t. w The total flow should equal the flow q required for the w-th trip demand. w The second formula represents the outflow endpoint t. w The total flow rate should be 0.

[0033] Step 4.2: Establish traffic balancing constraints for each node in the network;

[0034]

[0035] This constraint means that for each node in the network and for each travel demand, the flow into that node should equal the flow out of that node. This constraint ensures that the flow in the network will not suddenly disappear.

[0036] Step 4.3: Establish total flow constraints for each edge in the network;

[0037]

[0038] Among them, f ijpk For recovery scheme p, let p be the total traffic flow on road segment (i,j) when the k-th edge is restored. For recovery scheme p, let w be the flow rate of the w-th pair of travel demands on road segment (i,j) when the k-th edge is restored.

[0039] Step 4.4: Establish upper limit constraints on the flow for unrepaired edges;

[0040]

[0041] Step 4.5: Establish constraints on the relationship between the travel time and traffic for each edge in the network;

[0042]

[0043] Among them, t ijpk This indicates that when the total traffic flow on road segment (i,j) is f ijpk The passage time on that section of road.

[0044] Step 4.6: Establish the total travel time constraint for all traffic demands on the road network. This constraint is also the objective function in the user balanced traffic distribution model.

[0045]

[0046] Step 4.7: For T pk With flow f ijpk The nonlinear relationship is approximated by a secant;

[0047] Specifically: First, define a new parameter γ ij and two new variables y ijpk and g ijpk :

[0048]

[0049]

[0050]

[0051] Then equation (11) can be written as equation (15):

[0052]

[0053] Finally, we use a set of secant pairs y ijpk With g ijpkBy approximating the relationship, the following linear constraints can be obtained:

[0054]

[0055] Step 5: Model Solving – The proposed mixed-integer linear programming model is solved to efficiently obtain the optimal recovery scheme. This step relies on high-performance commercial solvers (such as CPLEX, Gurobi, etc.) to perform global optimization of the model.

[0056] (3) Advantages and benefits:

[0057] This invention can construct a mixed-integer linear programming (MILP) model based on the characteristics of real-world traffic distribution, enabling precise quantification and optimal recovery decisions for post-disaster road network capacity restoration. Compared to existing technologies, this invention has the following advantages:

[0058] ① Scientific nature and accuracy: This invention uses a user-balanced flow distribution model and resilience quantification index to accurately assess the impact of different recovery sequences on road network traffic efficiency, avoiding the blindness of traditional empirical methods and providing a scientific basis for decision-making.

[0059] ② Efficiency and practicality: This invention adopts a mixed-integer linear programming model, which transforms complex nonlinear problems into linear problems that can be solved efficiently. Combined with high-performance solvers (such as CPLEX and Gurobi), it can generate the optimal recovery scheme in a short time, making it suitable for practical applications of large-scale road networks.

[0060] ③ Comprehensive optimization and global perspective: This invention takes into account recovery time, traffic demand and traffic efficiency in a holistic manner, with the goal of maximizing resilience, and ensures that the road sections that most significantly improve the overall traffic capacity of the road network are repaired first, so as to quickly restore traffic function after the disaster and reduce social and economic losses. Attached Figure Description

[0061] Figure 1 This is a flowchart of a method for optimizing the restoration order of a damaged road network based on user-balanced flow distribution, with the goal of maximizing resilience, as described in this invention. Detailed Implementation

[0062] (1) Project Background

[0063] This implementation uses a small-scale road network in Beijing's urban area as the research object to simulate travel demand and obtain road network structure parameters and travel demand parameters. Based on this, it assumes that several road sections are damaged for some reason (such as heavy rain), and tests the above model and methods to obtain the optimal recovery sequence for the damaged road sections. This small-scale road network consists of 22 nodes and 68 edges, of which 6 are damaged road sections; there are a total of 182 pairs of traffic demand.

[0064] (2) Basic Information

[0065] The specific data on the damaged road sections are shown in Table 1 below:

[0066] Table 1 Specific data on damaged road sections

[0067] Serial Number starting point end Repair costs Repair time 1 10 16 1000 12 2 16 10 1000 12 3 16 17 2000 15 4 17 16 2000 15 5 17 19 1000 18 6 19 17 1000 18

[0068] (3) Optimize calculation results

[0069] Table 2 shows the optimization results of the restoration order.

[0070] Restore order starting point end Recovery effect (total passage time) 1 16 10 4179271 2 10 16 4120190 3 17 19 4088597 4 19 17 4054765 5 17 16 4022758 6 16 17 4003429

[0071] Table 2 shows the results of optimizing the recovery order using the proposed optimization method and model. The recovery effect represents the total travel time of the road network when the current road segment is restored. It can be seen that the proposed method fully complies with constraints such as flow conservation, user balance, and resilience, achieving optimal recovery order optimization for the damaged road network with the goal of maximizing the recovery effect (minimizing the total travel time), and successfully restoring the network.

[0072] (4) Comparative effect analysis

[0073] To verify the superiority of this invention, a comparative analysis was conducted with the "single-step optimal" method. The "single-step optimal" method is a logical and easily implemented recovery strategy. Its core idea is to find the edge that maximizes the reduction in travel time for the entire road network when recovering the first edge, and then use this as a basis to find another edge that maximizes the reduction in travel time for the entire road network among the remaining unrecovered edges, until all damaged road segments are recovered. While this recovery order yields a good result, it is not optimal.

[0074] Table 3 lists the results obtained by the intuitive "single-step optimal" method:

[0075] Table 3 Results obtained by the "single-step optimal" method

[0076] Restore order starting point end Recovery effect (total passage time) 1 17 19 4176363 2 19 17 4130256 3 17 16 4127834 4 16 17 4091081 5 16 10 4053750 6 10 16 4003429

[0077] As can be observed, Tables 2 and 3 respectively demonstrate the difference in the effectiveness of the "optimization calculation" and "single-step optimal" methods in terms of the restoration order. The restoration order in Table 2 reflects the correlation of multiple edges, thus exhibiting a stable and significant decreasing trend. In contrast, although the "single-step optimal" method in Table 3 selects the current optimal edge for restoration at each step, the reduction in total travel time is smaller. The final result is the same as in Table 2, but the effect of the intermediate process is worse. The reason why Table 2 performs better is that it considers the interrelationship between road segments in the network. In actual traffic networks, road segments do not exist independently; restoring one edge may have a chain reaction effect on the traffic conditions of other road segments. The method in Table 2, through global optimization, considers the restoration effect of multiple edges simultaneously, which can more effectively reduce the total travel time. The "single-step optimal" method only focuses on the current optimal solution, ignoring this correlation, resulting in local optimization failing to reach the global optimum. Therefore, the method in Table 2 performs better in overall travel time optimization, reflecting the importance of comprehensively considering network correlation.

[0078] In summary, this invention proposes an optimization method for the restoration order of damaged road networks based on user-balanced traffic allocation. By establishing a scientific resilience assessment framework and an efficient mixed-integer linear programming model, it solves the problems of insufficient decision-making basis and low efficiency in traditional post-disaster traffic restoration. This method innovatively quantifies road network resilience as the reciprocal of the product of restoration time and total travel time, and uses secant approximation to transform the complex nonlinear problem into an efficiently solvable linear programming problem. Combined with a high-performance optimization algorithm, it quickly generates the optimal restoration plan. Implementation results show that this method can accurately assess the impact of different restoration orders on road network traffic efficiency, prioritizing the repair of key road sections that most significantly improve overall traffic capacity, thereby significantly shortening traffic interruption time and improving the overall resilience of the road network. This invention provides a scientific and effective decision support tool for the rapid restoration of post-disaster transportation systems and has significant application value in emergency management and urban traffic planning.

[0079] This invention has the following core innovative features:

[0080] ① Resilience Assessment Framework: For the first time, the reciprocal of the product of recovery time and total travel time is used as a resilience indicator, quantifying the dynamic impact of recovery sequence on road network efficiency. A comprehensive resilience assessment method is constructed using a user-balanced flow distribution model, combined with road resistance functions and traffic demand, providing a theoretical basis for optimizing recovery sequence.

[0081] ② Multi-constraint linear optimization: Addressing the difficulty of solving traditional nonlinear models, this invention transforms complex nonlinear constraints into linear constraints through secant approximation and variable reconstruction, establishing an efficient mixed-integer linear programming model. This model not only preserves the actual physical meaning of the problem but also significantly improves solution efficiency and applicability.

[0082] ③ Dynamic Adaptability: An optimization strategy aimed at maximizing resilience is proposed. By defining decision variables and constraints, and coordinating factors such as traffic demand, node balance, and travel impedance, the optimality of the recovery plan is ensured globally. This method overcomes the limitations of traditional static prioritization, achieving dynamic and adaptive road network recovery planning, and providing an innovative solution for post-disaster traffic management.

Claims

1. A method for optimizing the restoration order of damaged road networks based on user-balanced traffic distribution with the objective of maximizing resilience, comprising the following steps: (1) Construction of a framework for assessing the resilience of damaged road networks: Based on the comprehensive assessment of the damage to the traffic network topology and traffic demand after the disaster, and based on the user balance distribution model, the assessment method for the resilience of damaged road networks is determined. (1.1) First, the relationship between the travel time t of the road and the flow rate f on the road segment is expressed by the following road resistance function; (1.2) Next, the objective function for user equilibrium assignment, i.e., the integral of travel time over flow, f ij is the flow on link (i, j). (1.3) Assume there are k damaged links, each with a required recovery time of τ k For each damaged link recovered, the total travel time for all traffic demands on the network should decrease, so the total travel time T k is multiplied by the cumulative inverse of the required recovery time for each damaged link recovered as the recovery resilience R of the network: (2) Parameter definition for the scenario of optimizing the resilience of damaged road networks The traffic network is mathematically abstracted as a connected graph G = (V, E), where i, j ∈ V, are the indices of road nodes, i, j = 1, 2, …, |V|, and |·| represents the total number of elements in the set. Nodes correspond to intersections and key locations in the traffic network with traffic demand, such as schools, factories, hospitals, banks, office buildings, and residential areas. The set E' represents the set of damaged roads, E' ⊂ E; the set K represents the set of repair sequence numbers for repairing damaged road sections, k ∈ K, k = 1, 2, …, |K|, and |K| = |E'|; the set P represents the set of repair plans, p ∈ P, where each element corresponds to an arrangement of repair sequences for damaged road sections; the parameter h... ijpk The value is either 0 or 1, indicating whether edge (i, j) has been restored when the restoration scheme p reaches the k-th edge. If it has been restored, h... ijpk = 1, otherwise h ijpk = 0; correspondingly, τ pk This represents the recovery time required to recover the k-th edge for recovery scheme p; each edge (i, j) ∈ E in the connected graph G represents a road in the traffic network, t 0ij c represents the free-flow time of this edge, that is, the shortest time to traverse this edge; ij Let t represent the capacity of edge (i, j), which is the maximum number of vehicles that can pass through this road within a certain time period; for the damaged road segment, t 0ij and c ij These represent the free-flow time and capacity of the road section after repair; α is a parameter in the road resistance function, with a value of 0.15; This is a parameter in the road resistance function, with a value of 4; The parameters introduced when approximating the path resistance function by a secant are α, β, and c. ij Parameters were calculated together. and These represent the slope and intercept of the secant line introduced when approximating the path resistance function using the secant line, respectively. Let d represent the slope of the secant line when approximating edge (i, j); Let represent the intercept of the d-th secant when approximating edge (i, j); W represents the set of travel demands, w∈W, w = 1, 2, …, |W|; (s w , t w ) represent the origin and destination of the w-th travel request, respectively. w , t w ∈V; parameter q w This represents the flow rate of the w-th travel demand; M is a large number. (3) Establish a mixed integer linear programming model for the restoration order of damaged road networks with the goal of maximizing resilience. The decision variables are the flow rate on each road segment, the resilience corresponding to each restoration scheme, and the integral time obtained by each restoration step. The optimization objective is to maximize the restoration resilience, which is equivalent to minimizing the reciprocal of the restoration resilience. Define the decision variables for optimizing the restoration order of the damaged road network with the objective of minimizing the inverse of the recovery resilience; : non-negative continuous variable representing the flow of the wth pair of travel demand on link (i, j) when recovering to the kth edge for recovery scheme p; f ijpk : non-negative continuous variable representing the total flow on link (i, j) that travels to the kth edge for recovery scheme p; t ijpk : non-negative continuous variable representing the travel time on link (i, j) when the total flow f ijpk passes on this link. y ijpk : non-negative continuous variable, introduced as an intermediate variable when linearizing the non-linear impedance function; g ijpk : non-negative continuous variable, introduced as an intermediate variable when linearizing the non-linear impedance function; T pk : non-negative continuous variable, representing the total travel time of all flow demands on the road network when recovering to the kth edge for recovery scheme p; B p : non-negative continuous variable representing the inverse of the resilience to recovery scheme p; (3.1) Establish an objective function for optimizing the restoration order of the damaged road network with the goal of maximizing resilience; wherein B p T is the inverse of the recovery robustness of recovery scheme p pk denotes the total travel time of all traffic demands on the road network when recovering to the kth edge for recovery scheme p, reflecting the travel efficiency of the road network pk denotes the recovery time required to recover the kth edge for recovery scheme p, and the minimum of their product reflects the goal of quickly improving the travel efficiency of the road network (4) Constrain the model by taking into account traffic demand, traffic balance and travel impedance; (4.1) Establish travel demand constraints; (4.2) Establish traffic balancing constraints for each node in the network; (4.3) Establish total flow constraints for each edge in the network; (4.4) Establish upper limit constraints on the flow of unrepaired edges; (4.5) Establish constraints on the relationship between the travel time and traffic of each edge in the network; (4.6) Establish the total travel time constraint for all traffic demands on the road network, which is also the objective function in the user balanced traffic distribution model; (4.7) For T pk a non-linear relationship with the flow f ijpk is approximated by a secant, first define a new parameter and two new variables y ijpk and g ijpk ; (4.8) T pk is written as; (4.9) Approximating the relationship of y ijpk with g ijpk the following linear constraints are obtained; (5) Model Solving: The proposed mixed integer linear programming model is solved to efficiently obtain the best recovery scheme; this step relies on the high-performance commercial solver CPLEX to perform global optimization of the model.

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