Bulk dry bulk cargo hinterland multimodal transport network and inventory joint optimization method considering uncertainty

Abstract

The present disclosure relates to a bulk dry cargo hinterland multimodal transport network and inventory joint optimization method considering uncertainty, which is used to solve the problems of weak network risk resistance, low heuristic algorithm solution quality and slow precise algorithm solution speed existing in the existing multimodal transport network and inventory joint optimization technical scheme. The present scheme presets low-cost nodes around the core hub as backup hubs, establishes a hub dynamic replacement mechanism based on "flat and urgent operation", and designs a Benders-CG solving method based on multiple scenarios, effectively solving the contradiction between network resilience, cost control and solving efficiency.

Description

Technical Field

[0001] This disclosure relates to freight scheduling, and more particularly to a joint optimization method for multimodal transport networks and inventory in the hinterland of bulk dry cargo, taking into account uncertainties. Background Technology

[0002] Bulk dry cargoes (such as coal, iron ore, and grain) are fundamental materials of the national economy, characterized by large single-batch volumes, long transportation distances, and relatively low unit values. The stability of their supply chains directly affects national energy security, industrial chain stability, and grain reserve security. Statistics show that in 2023, my country imported approximately 1.179 billion tons of iron ore, 474 million tons of coal, and 162 million tons of grain. This massive flow of goods places stringent demands on logistics cost control and transportation efficiency. Therefore, multimodal transport, combining the low-cost advantages of water transport and rail transport with the mobility of road transport, has become the mainstream mode for the collection and distribution of imported bulk cargoes from coastal areas to the hinterland.

[0003] Hinterland bulk dry cargo multimodal transport systems typically exhibit a hub-and-spoke network structure (such as...) Figure 1 (As shown). This network uses coastal seaports as entry gateways, transits through inland transshipment hubs (such as inland river ports, railway marshalling yards, and large storage yards), and ultimately radiates to demand terminals deep in the hinterland, such as steel bases or thermal power plants. Specifically, after goods enter through the gateways, they mainly choose three types of transportation routes based on cost-effectiveness: first, a multi-level transshipment route involving deep hinterland transportation, namely "seaport gateway → transshipment hub → transshipment hub → demand terminal"; second, a standard intermodal transport route, namely "seaport gateway → transshipment hub → demand terminal"; and third, a direct route, namely "seaport gateway → demand terminal," covering some short distances. The nodes are organically interwoven with railway trunk lines, inland waterways, and highway networks. In actual operation, this multi-level hub-and-spoke structure means that goods often need to undergo complex cross-modal transport transshipments, making route decisions and flow allocation in the network highly coupled. In addition, the ports and inland storage yards in this system are not only transportation mode conversion hubs, but also undertake the key function of inventory buffering. They effectively solve the problem of supply and demand mismatch in time and space by utilizing their own storage capacity. However, under the existing model, these valuable storage resources are often passively used only for basic heap storage, lacking proactive planning and joint optimization from the perspective of network-wide resilience.

[0004] Based on the aforementioned characteristics of multimodal transport networks, simply optimizing transport routes is insufficient to meet the requirement of optimal overall system efficiency in large-scale dry bulk cargo circulation. This is because there is a natural coupling between transportation and inventory: transportation facilitates the spatial transfer of goods, while inventory plays a crucial role in mitigating supply and demand fluctuations and avoiding stockout risks. Inventory levels directly determine whether the supply of dry bulk cargo can be maintained in the face of disturbances, thus ensuring uninterrupted subsequent transportation tasks; conversely, efficient transportation scheduling can effectively reduce the risk of inventory backlog. Therefore, only by jointly optimizing the multimodal transport network and inventory for dry bulk cargo can the optimal trade-off between supply assurance and operating costs be found in a dynamically changing environment. This is the so-called joint optimization problem of the multimodal transport network and inventory for dry bulk cargo.

[0005] Specifically, this problem requires decision-makers to conduct layered collaborative planning, with core decisions typically encompassing two levels: First, the strategic level of network configuration, which determines the site selection for the multimodal transport network infrastructure. Its main task is to select inland railway stations, ports, or storage yards with transshipment hub functions from numerous potential nodes, thereby building the framework for the entire multimodal transport system. Second, the operational level of dynamic transport and inventory coordination, which involves scheduling decisions within a given network structure. This requires refined control of traffic allocation and inventory management in each time period. On the one hand, it involves deciding on specific cargo transport volumes and directions for rail, road, waterway, and their combined routes; on the other hand, it involves formulating inventory levels and replenishment strategies for each hub at the end of each time period. This ensures both the accessibility of bulk dry bulk cargo and sufficient inventory levels to cope with potential shortages, achieving optimal matching between bulk dry bulk transport and storage in a dynamic environment. In summary, the essence of this problem is to find a comprehensive site selection and scheduling scheme that minimizes the total cost of the entire network, while meeting end-user demand and physical constraints such as transport capacity and storage space.

[0006] However, the complexity of the real-world environment goes far beyond this. In actual operation, multimodal transport networks in the hinterland of dry bulk cargo often face the dual disturbances of supply and capacity uncertainty. Supply uncertainty primarily stems from trade and transportation disruptions caused by international geopolitical games, as well as production fluctuations in upstream mines due to seasonal factors, policies, or production accidents. Capacity uncertainty, on the other hand, frequently occurs during hinterland transport, such as capacity reductions caused by extreme weather, sudden waterway congestion, or planned railway maintenance. This dual uncertainty, resulting from the superposition of macro-geopolitical and micro-operational factors, makes established networks highly susceptible to disruptions, posing a severe challenge to the resilience and anti-interference capabilities of multimodal transport systems.

[0007] The shortcomings of existing multimodal transport networks and inventory optimization technologies for bulk dry cargo can be summarized in the following three aspects: (1) Weak network resilience: Existing optimization schemes are severely lacking in robustness when dealing with complex environments, mainly in two key dimensions: First, the consideration of uncertainty is too one-sided. Existing technologies usually only consider uncertainty in a single dimension, ignoring the superposition of supply and capacity uncertainties that often occur in reality, resulting in insufficient resilience of the model-generated schemes against real-world risks. Second, there is a lack of dynamic hub replacement mechanisms. The existing network structure design is too rigid. Once a core hub suffers from insufficient capacity or functional paralysis due to a sudden event, the system lacks a pre-set backup hub for dynamic replacement and rescheduling. This static network without backup plans cannot achieve flexible switching between nodes in times of crisis, thus causing local congestion to quickly evolve into a global supply chain disruption.

[0008] (2) The quality of heuristic algorithms is not high: Although metaheuristic algorithms such as genetic algorithms and particle swarm optimization can quickly handle large-scale network problems, they are essentially approximate search based on probability. Due to the lack of mathematical proof of optimality and lower bound evaluation, these algorithms are very prone to falling into local optimum traps, resulting in uncontrollable deviations between the solution and the theoretical optimal solution.

[0009] (3) Slow solution speed of exact algorithms: Although exact algorithms aim to find the theoretically optimal solution, existing technologies face severe computational bottlenecks when dealing with large-scale two-stage stochastic mixed integer programming problems such as bulk dry cargo multimodal transport networks. This not only leads to huge memory consumption during the computation process, which can easily cause memory overflow, but also makes it difficult to converge effectively within the time window allowed by the project, forcing decision-makers to accept a suboptimal intermediate solution after a long wait. Summary of the Invention

[0010] Based on this, this disclosure proposes a joint optimization method for multimodal transport networks and inventory in the hinterland of bulk dry cargo, which takes into account uncertainties. This method has a highly resilient network planning and scheduling scheme, and can effectively mitigate the risk of supply and demand disruption by flexibly adjusting the network structure and proactively buffering inventory when facing these uncertain shocks, thus ensuring a continuous and stable supply.

[0011] Firstly, this disclosure proposes a joint optimization method for multimodal transport networks and inventory considering uncertainties. Based on the multimodal transport network and inventory, an expected comprehensive cost function and constraints are constructed. The expected comprehensive cost includes transportation costs, warehousing costs, stockout costs, and the activation cost of temporarily activating a backup hub as the core hub. The constraints include constraints based on a dynamic hub replacement mechanism, which involves pre-setting backup hubs around the core hub, and these backup hubs are capable of warehousing and transshipment conversion. Under these constraints, Benders decomposition and column generation are used to solve for decision variables that minimize the expected comprehensive cost. These decision variables include: a core hub selection variable. : Select a node j As a core hub, , For the set of potential core hubs; alternative hub selection variables : Select a node i As a core hub j backup hub , W For the set of potential backup hubs; path selection variables Within period t, the scenario Should a transportation method be used? From node To the node transportation, , , , For a set of nodes, For the collection of transportation modes, For a periodic set, A collection of scenarios; backup hub activation variable In the scene Next, cycle Internal backup hub Whether it will be activated as a temporary core hub ; Flow allocation variables In the cycle Inside, scene Next node To the node By transportation Freight volume of transportation , , , Inventory quantity variable Scene Next, node In the cycle Ending inventory , DFor the set of demand points; for the stockout quantity variable Scene Next, node In the cycle Stockouts at the end of the period, .

[0012] Secondly, this disclosure proposes a computer-readable storage medium storing a computer program that can be loaded by a processor and execute any of the methods of this disclosure.

[0013] Thirdly, this disclosure proposes a multimodal transport network and inventory joint optimization system considering uncertainty, comprising: a model building module, configured to construct an expected comprehensive cost function and constraints based on the multimodal transport network and inventory, wherein the expected comprehensive cost includes transportation cost, warehousing cost, stockout cost, and activation cost of temporarily activating a backup hub as a core hub; the constraints include constraints based on a dynamic hub replacement mechanism, wherein the dynamic hub replacement mechanism involves pre-setting backup hubs around the core hub, and the backup hubs are capable of warehousing and transshipment conversion; and a solution module, configured to solve for the decision variables that minimize the expected comprehensive cost under the constraints, wherein the decision variables include: a core hub selection variable. : Select a node j As a core hub, , For the set of potential core hubs; alternative hub selection variables : Select a node i As a core hub j backup hub , W For the set of potential backup hubs; path selection variables Within period t, the scenario Should a transportation method be used? From node To the node transportation, , , , For a set of nodes, For the collection of transportation modes, For a periodic set, A collection of scenarios; backup hub activation variable In the scene Next, cycle Internal backup hub Whether it will be activated as a temporary core hub ; Flow allocation variables In the cycle Inside, scene Next node To the node By transportation Freight volume of transportation , , , Inventory quantity variable Scene Next, node In the cycle Ending inventory , D For the set of demand points; for the stockout quantity variable Scene Next, node In the cycle Stockouts at the end of the period, .

[0014] Based on the above technical solutions, this disclosure effectively resolves the contradiction between network resilience, cost control, and solution efficiency by constructing a hub dynamic replacement mechanism and a Benders-CG hybrid solution algorithm. Specific advantages are as follows: ① The network planning method, combining consideration of dual uncertainties with the hub dynamic replacement mechanism, can quickly achieve path reconstruction and inventory buffering under sudden risks, helping to ensure the continuous and stable supply of bulk dry goods in extreme scenarios; ② The adoption of a "normal and emergency combined" backup hub operation mode can reduce costs by storing inventory under normal conditions and ensure transport capacity by switching functions in emergencies, improving resource utilization and balancing high supply chain resilience with optimal economic efficiency throughout the entire lifecycle; ③ The Benders-CG algorithm, combining the advantages of Benders and column generation, can provide an accurate optimal solution to the problem in a short time. This method has strong universal applicability and can meet the needs of rapid solution for problems of different scales. Attached Figure Description

[0015] To more clearly illustrate the technical solutions in the embodiments of this application, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0016] Figure 1 This is a schematic diagram of the existing design for a multimodal transport network for bulk dry cargo in the hinterland.

[0017] Figure 2 A schematic diagram of the multimodal transport network in the hinterland of bulk dry cargo.

[0018] Figure 3 This is a schematic diagram of the hub dynamic replacement mechanism.

[0019] Figure 4 A schematic diagram of the Benders-CG solution process. Detailed Implementation

[0020] To address the shortcomings of existing technologies, such as insufficient network resilience due to neglecting the uncertainties of both supply and transport capacity, and the difficulty of balancing solution quality and computational efficiency in large-scale stochastic programming models, this patent proposes a joint optimization method for bulk dry bulk multimodal transport networks and inventory that considers these dual uncertainties. This method aims to achieve high resilience and efficient decision-making in bulk dry bulk multimodal transport systems within complex and dynamic operating environments through technological breakthroughs at two core levels.

[0021] First, this patent introduces a dynamic hub backup mechanism. Unlike the fixed network structure in existing technologies, this patent pre-determines low-cost nodes around the core hub as backup hubs during the strategic site selection phase and establishes dynamic switching rules based on "normal and emergency operations." Under normal operation, the relatively low warehousing cost advantage of backup hubs is utilized to serve as auxiliary warehouses for the core hub, reducing the overall inventory holding cost of the system through diverted storage. In emergency scenarios, when the core hub experiences insufficient or interrupted transport capacity due to natural disasters or accidents, the system can automatically activate backup hubs to assume transshipment functions and reconstruct transport routes. By using inventory as a buffer resource to cope with the risk of supply chain disruptions, the system effectively mitigates the impact. When the supply of bulk cargo at seaports is insufficient due to geopolitical conflicts, extreme weather, or other reasons, the inventory stored at backup hubs and other nodes can be quickly converted into emergency supply, ensuring the continuity of production at end-user demand points and thus avoiding the risk of production stoppages caused by supply disruptions. This mechanism transforms the network structure from static rigidity to dynamic elasticity, effectively mitigating the risk of global collapse caused by local failures and ensuring a continuous and stable supply of bulk dry cargo under extreme scenarios.

[0022] Secondly, this patent develops a Benders-CG algorithm based on multiple scenarios. Faced with a large number of decision variables and constraints in the optimization model of bulk dry cargo multimodal transport networks, traditional commercial solvers struggle to solve directly, while the classic Benders decomposition algorithm suffers from bottlenecks such as relaxed cut planes and slow convergence. This invention proposes a hybrid solution acceleration strategy combining Benders decomposition and column generation (CG). Benders decomposition is used to break down the large-scale mixed-integer programming problem into a main problem and sub-problems to reduce model dimensionality. Given that the number of potential binary variables in the main problem grows exponentially with network size, and that a large number of invalid Benders cuts are generated in each scenario, this patent constructs the main problem as a restricted main problem and introduces column generation technology for acceleration. In the solution process, only a small number of solutions are considered in the initial stage. Subsequently, by constructing a pricing sub-problem, key cost-saving solution columns are dynamically identified and added to the main problem, thus avoiding the calculation of massive invalid solutions. Simultaneously, using the Benders cut planes fed back from solving the sub-problems in each scenario, constraints are continuously added to the main problem to accurately approximate the expected future operating cost. This accelerated solution algorithm, which uses column generation to dynamically select solutions to the main problem and Benders decomposition to provide feedback on sub-problem scenario information, effectively solves the computational challenge of having too many variables and loose constraints in large-scale mixed integer programming. It significantly improves computational efficiency while ensuring the attainment of the global optimal solution.

[0023] This technology can solve the following three problems.

[0024] ① This solution addresses the problem of existing static, rigid networks being prone to chain disruptions and paralysis under dual uncertainties. By introducing a dynamic hub replacement mechanism, the network gains self-healing resilience. When faced with dual shocks from both the supply and transportation capacity sides, the system can detect and activate backup hubs in real time to reconstruct transportation routes, utilize inventory buffers to resolve supply and demand gaps, effectively prevent the spread of local risks to the overall system, and ensure the resilience and stable operation of the bulk dry cargo supply chain under extreme scenarios.

[0025] ② This solution addresses the issues of high construction costs and low utilization rates of traditional emergency facilities. Traditional high-resilience networks often rely on building a large number of idle redundant facilities, resulting in extremely high sunk costs. An innovative "normal / emergency combined" operation model is adopted, selecting low-cost surrounding nodes as backup hubs, maintaining safety stock under normal conditions, and removing constraints during emergencies. This design achieves dual optimization of supply chain security and economy, ensuring maximum overall benefits at different operational stages.

[0026] ③ This solves the problem that large-scale stochastic mixed-integer programming models struggle to find the global optimum in a short time. Addressing the curse of dimensionality caused by massive variables and uncertainties in bulk dry cargo network optimization, a Benders-CG hybrid acceleration algorithm is proposed, effectively overcoming the computational bottlenecks caused by excessive variables and relaxed constraints. This enables decision-makers to obtain a mathematically proven global optimum in a short time.

[0027] The following provides a clear and complete description of how the technical solution of this case is implemented. Obviously, the described implementation methods are only a part of the implementation methods of this case, and not all of them. Based on the implementation methods in this case, all other implementation methods obtained by those skilled in the art without inventive effort are within the scope of protection of this application.

[0028] (a) Multimodal transport network in the hinterland of bulk dry cargo Consider a case such as Figure 2 The diagram illustrates a multimodal transport network for bulk dry cargo, comprised of supply points, potential inland transshipment hubs, and hinterland demand points. Seaport gateways (supply points) refer to coastal ports that handle the unloading of international bulk cargo and its distribution to inland areas, such as Ningbo-Zhoushan Port and Shanghai Port. Potential inland transshipment hubs are candidate nodes located in the inland hinterland, possessing the capability to switch between two or more modes of transport and warehousing capacity. These can be further divided into two categories: potential core hubs and potential backup hubs. Core hubs typically refer to large logistics nodes occupying key locations in the transportation network, with mature infrastructure, and capable of handling major traffic transshipment and distribution tasks under normal circumstances, such as Wuhan and Zhengzhou. Backup hubs typically refer to auxiliary nodes located within the radiation range of core hubs, providing flexible buffers and emergency backup functions; their warehousing costs are usually lower than those of core hubs, such as Ezhou. Demand points typically refer to the final destination nodes in the supply chain that have actual production consumption needs for bulk dry cargo, such as hinterland steel mills and power plants. Among all nodes, the core hub, backup hub, and demand point have warehousing capabilities. Among them, the backup hub has the highest storage capacity, the lowest unit storage price, and safety stock.

[0029] During the planning period, several core hubs will be selected from potential core hub points to undertake the main transshipment and distribution functions. For each selected core hub, several potential backup hubs will be selected from potential backup hubs to provide auxiliary storage support and emergency backup in case of emergencies. Both will operate in tandem to ensure a continuous and stable supply of bulk dry cargo from supply points to demand points. Considering the economies of scale requirements of bulk cargo transportation and to reduce the complexity of network scheduling, each supply point or demand point will be served by only a specific number of core hub points.

[0030] After the multimodal transport network framework is completed, the system will enter the specific operation and scheduling phase. Given that imported bulk dry cargo mainly flows from coastal areas to inland regions, it is necessary to comprehensively plan multimodal transport routes, cargo volume allocation, and inventory control strategies at each node based on the actual supply and demand situation at different times. Under normal circumstances with sufficient transport capacity, bulk dry cargo enters the country from the supply point and flows to the hinterland through three routes: ① Route 1: Efficient Direct Transportation For large demand points that are close to the supply point or have dedicated transportation channels, bulk dry cargo can be delivered directly from the supply point to the demand point without going through any transit hubs, thereby reducing transit and loading / unloading links and reducing logistics losses.

[0031] ② Route Two: Core Hub Distribution To address the dispersed demand in the deep inland hinterland, goods are transported via trunk lines to core hubs, where they are integrated and transshipped using the economies of scale of the hubs, and then distributed to various demand points.

[0032] ③ Path Three: Adjustment of Reserve Stock Some goods are directly transported to the backup hub. Under normal circumstances, the backup hub mainly uses its low-cost advantage to serve as a strategic reservoir, receiving and storing some of the cargo flow from the supply points, and coordinating with the core hub to supplement the supply to the demand points according to the supply and demand gaps in the upstream and downstream, thereby balancing the overall network inventory costs.

[0033] The combined effects of rising trade protectionism, intensified anti-globalization trends, and sudden public health emergencies have created significant uncertainty and vulnerability in the external supply environment for my country's bulk dry cargo. Furthermore, the capacity of the domestic hinterland's distribution system also faces severe uncertainty. Inland waterway shipping is highly susceptible to seasonal low water levels, flood season flow restrictions, and severe weather such as typhoons, leading to navigation disruptions; rail transport is constrained by planned concentrated maintenance and capacity reductions due to sudden equipment failures. The combination of these natural and man-made factors often results in localized capacity shortages, seriously threatening the continuity of the supply chain. In summary, the combined risks of supply-side disruptions caused by the international environment and capacity fluctuations caused by the domestic operating environment constitute the severe challenge of dual uncertainties in supply and capacity that the current bulk dry cargo multimodal transport network must confront.

[0034] When the system encounters uncertainties in both capacity and supply, the network will mitigate risks through dynamic hub replacement mechanisms and inventory adjustments. If direct routes are disrupted due to natural disasters or maintenance, the system will automatically reroute direct traffic to the nearest core or backup hub for transfer. If the main channel of a core hub becomes blocked, the system will immediately activate the transfer function of the associated backup hub, making it a core hub, as detailed below. Figure 3As shown. At this point, backup hubs can connect with supply points, other core hubs, or demand points to ensure uninterrupted cargo flow and that inventory levels are no longer constrained by safety stock. If the capacity of core hubs is severely depleted, multiple backup hubs can be activated to assume transshipment responsibilities. If insufficient supply or capacity prevents demand points from being met, the system will prioritize consuming the locally accumulated inventory at the demand point, while simultaneously allocating previously accumulated inventory from core and backup hubs as emergency supply sources to transport to the demand point to ensure production in the hinterland.

[0035] The entire planning period was divided into Each cycle, therefore decision-makers need to Within discrete periods, joint decisions are made regarding hub location, inventory levels, and multimodal transport route traffic for different potential risk scenarios to ensure that demand is met while achieving the global optimum of the expected comprehensive cost within the planning period. The expected comprehensive cost consists of four parts: transportation cost, inventory cost, stockout cost, and temporary hub activation cost.

[0036] Due to differences in transportation methods and distances, the unit transportation cost between nodes also varies. However, the overall transportation cost consists of the following four parts: transportation cost from supply point to demand point, transportation cost from supply point to transit hub (core hub, backup hub), transportation cost from transit hub to transit hub, and transportation cost from transit hub to demand point.

[0037] Inventory cost refers to the holding expenses incurred at the end of each cycle for retaining bulk dry cargo in core hubs, backup hubs, and demand points to adjust for supply fluctuations or cope with sudden disruptions. This cost is positively correlated with the current inventory holding level and unit warehousing fee rate at each node.

[0038] Stockout costs refer to the punitive charges imposed on unmet cargo gaps when the effective supply of a multimodal transport network cannot fully meet the demand of the end users or when proactive stockouts are chosen due to high freight costs.

[0039] Temporary hub activation cost: refers to the one-time activation fee or additional operating cost incurred when the emergency response mechanism is triggered in an emergency scenario, and the backup hub is switched from a normal static auxiliary storage function to an emergency transit and distribution function.

[0040] (II) Modeling based on the multimodal transport network diagram of the hinterland of bulk dry cargo (2.1) Basic Assumptions ① The upper limit of storage capacity of core hubs, backup hubs and demand points is known, the upper limit of transport capacity of different modes of transport between nodes is known, and it remains stable within the planning period.

[0041] ② Only one mode of transportation, namely highway, railway, or waterway, can be selected between any two nodes for cargo transportation. The unit transportation cost is only related to the node pair and the mode of transportation and does not fluctuate over time.

[0042] ③At the beginning of the planning period, all nodes have zero inventory, and the unit inventory cost is fixed and known.

[0043] ④ The probability of each scenario is known and the sum of the probabilities of all scenarios is 1. Different scenarios are independent of each other and do not affect each other.

[0044] ⑤ Fixed costs such as the selection of core hubs and the selection and allocation of backup hubs are not considered separately, but are allocated to inventory costs.

[0045] ⑥ There is sufficient time to transport goods within each cycle, and there is no cross-cycle transportation.

[0046] ⑦ The temporary activation cost of the backup hub is a one-time fixed cost, which is unrelated to the activation duration and the volume of transshipped goods.

[0047] (2.2) Set settings S Supply point collection ; D : Set of demand points ; H : A collection of potential core hubs ; W Potential backup hub set ; V : Node set ; M The term "transportation mode" refers to a collection of transportation methods, including road transport, rail transport, and water transport. ; T Periodic sets ; The collection of scenarios may cause changes in the supply volume at supply points or the transportation capacity between nodes. .

[0048] (2.3) Parameter settings : Using transportation methods m via arc ( i , j The unit transportation cost, ; :nodej Unit inventory cost ; Temporary hub i The activation cost ; :node j Inventory limit, ; Scene Next, cycle t inner arc ( i, j The upper limit of the transport capacity of mode m on the transport route. ; Unit penalty cost for stockouts at demand points; The maximum number of hub points that each supply point is allowed to directly connect to; The maximum number of hub points that can be directly connected to each demand point; Demand Points j In the cycle t Domestic demand ; Supply point i In the cycle t Internal supply ; Safety stock of backup hubs ; : An extremely large number; Scene The probability of occurrence, and .

[0049] (2.4) Decision variables Within period t, the scenario Arc ( i , j By mode of transport m Freight volume of transportation , , , ; : 0-1 variables, within a period t, scenario Should a transportation method be used? m In arc ( i ,j )transportation, , , ; : 0-1 variable, whether to select a node j As a core hub, ; : 0-1 variable, whether to select a node i As a core hub j backup hub ; Scene Next, node j In the cycle t Ending inventory ; Scene Next, demand points j In the cycle t Stockouts at the end of the period, ; In the scene Next, cycle t Internal backup hub i Whether it will be activated as a temporary core hub .

[0050] (2.5) Model building Based on the problems described by the multimodal transport network in the hinterland of bulk dry cargo, the following model is constructed.

[0051] Objective function:

[0052] Constraints:

[0053] The objective function (1) represents minimizing the expected comprehensive cost, including transportation costs, warehousing costs, stockout costs, and the activation of backup hubs as temporary core hubs. Constraint (2) stipulates that the total outflow from a supply point in each time period must not exceed the maximum supply capacity of that node in the corresponding scenario. Constraints (3) and (4) are flow conservation constraints for the core hub and backup hub, respectively, ensuring that the inflow plus the beginning inventory equals the outflow plus the ending inventory. Constraint (5) is the flow balance and stockout definition constraint for demand points, calculating the stockout amount when the sum of inflow and inventory fails to meet demand. Constraint (6) ensures that the holdings of the core hub, backup hub, and demand point do not exceed their maximum physical storage capacity. Constraint (7) stipulates that when a backup hub is not activated, it acts as a warehouse and must maintain a certain minimum inventory level. Constraint (8) ensures that only backup hubs allocated to the core hub are allowed to hold inventory. Constraint (9) restricts the freight flow on each path from exceeding the upper limit of the path's capacity, and allows flow only when the path is open. Constraint (10) restricts the upper limit of the number of transportation paths that each supply point connects to. Constraint (11) limits the maximum number of inbound transport routes that each demand point can accept. Constraint (12) ensures that only one mode of transport can be selected to open a route between two nodes. Constraints (13)-(14) ensure that only selected core hubs can establish inbound connections with supply points and outbound connections with demand points. Constraint (15) mandates that each selected core hub must be equipped with a backup hub. Constraint (16) stipulates that each backup hub can serve at most one core hub. Constraints (17) and (18) ensure that transport routes between them exist only when an attribution relationship is established or a backup hub is activated. Constraints (19) and (20) together guarantee that transport relationships can only exist between selected core hubs and cannot be established with unselected hubs. Constraint (21) ensures that backup hubs can only be assigned to core hubs that have already been selected. Constraint (22) stipulates that the stockout quantity must not exceed the total demand for the current period under any circumstances. Constraints (23) and (24), together with constraints (17) and (18), ensure that direct transport with nodes other than the core hub is permitted only when the backup hub is activated. Constraint (25) ensures that only backup hubs that have been assigned affiliations are eligible for activation. Constraints (26) and (27) declare the domain of the variables. Constraint (28) sets the initial inventory of all nodes to zero at the start of the planning period.

[0054] Constraint (7) is a nonlinear constraint. After linearization, the formula is as follows:

[0055] The simplified version is as follows:

[0056] The aforementioned model establishes a joint optimization model for the multimodal transport network and inventory of bulk dry cargo, considering the uncertainties of both supply and transport capacity. By solving for the decision variables, it enables multi-decision coordination in hub selection, route planning, inventory control, and emergency response. The model construction incorporates a dynamic hub backup mechanism that combines normal and emergency operations. By pre-setting backup hubs and switching between normal warehousing and emergency transshipment functions, the model enhances the resilience of the multimodal transport network in the hinterland of bulk dry cargo under dual uncertainties. Through a dual-state constraint rule for backup hub inventory, a safety stock is maintained under normal conditions, and the constraint is lifted during emergencies, achieving a dynamic balance between cost control and risk buffering of inventory resources.

[0057] (III) Model Solving This invention designs an exact solution algorithm combining Benders decomposition and column generation (CG), namely the Benders-CG algorithm, which is a collaborative algorithm with inner and outer loops. The outer loop uses the Benders decomposition strategy to decompose the original problem into a restricted master problem (RMP) and a dual subproblem (DSP). By solving the DSP, the upper bound is updated, and feasible cuts or optimal cuts are generated based on dual information to continuously tighten the constraints of the master problem. The inner loop embeds a column generation mechanism in the RMP. By iteratively solving the relaxed RMP and pricing subproblem (SP-CG), the key solution column with negative test numbers is dynamically screened. Then, the integer RMP is solved to obtain the current optimal location and network construction scheme and the lower bound is updated until the difference between the upper and lower bounds converges to the preset precision, thus achieving a globally accurate solution to the problem.

[0058] (3.1) Benders decomposition The Benders decomposition involves breaking down the original problem into a main problem and subproblems based on the characteristics of the decision variables. In the main problem (MP), all decision variables are integer variables (0-1), i.e., the core pivot selection variable (…). ), Backup hub selection variables ( ), path selection variables ( ) and backup hub activation variables ( The subproblem (SP) contains only continuous variables, namely flow allocation variables. ), inventory quantity variable ( ) and stockout quantity variable ( ).

[0059] Suppose a set of integer solutions is known. The original problem is then decomposed according to the above decomposition rules. At this point, the original constraints of the subproblem SP include constraints (2)-(9) and constraint (22), where, Substitute constraints (7)-(9). To facilitate the derivation of the dual subproblem (DSP), the subproblem SP is standardized, that is, except for constraints (3)-(5), the remaining constraints are uniformly written as "" through rearranging terms or changing signs. "form.

[0060] The subproblems SP obtained after decomposing and standardizing the original problem are as follows: Objective function:

[0061] Constraints:

[0062] The physical meaning of SP refers to the situation where, given a decision... Next, find continuous variables. The optimal configuration. If SP is feasible, then the following can be obtained. optimal solution and optimal value If SP is not feasible, then the decision is... If the constraints of a subproblem cannot be satisfied, it needs to be removed by adding a feasible cut to the main problem. Since SP is a linear programming problem, it has a dual problem. Therefore, information can be fed back to the main problem through the solution of the dual problem.

[0063] For each subproblem constraint, a dual variable is introduced, defined as follows: The dual variable of constraint (32), , ; : The dual variable of constraint (33), ; : The dual variable of constraint (34), ; The dual variable of constraint (35), ; The dual variable of constraint (36), ; The dual variable of constraint (37), ; : The dual variable of constraint (38), ; The dual variable of constraint (39), , ; The dual variable of constraint (40), .

[0064] Based on the subproblem, the DSP derivation is as follows: Objective function:

[0065] Constraints:

[0066] when t When it is the last cycle, it no longer exists. t +1 cycle, meaning it does not exist. , and Variables. At this point, constraints (52)-(54) are rewritten as follows:

[0067] To effectively avoid the problem caused by a surge in the number of scenarios, the DSP can be broken down into... The dual problem (DSP) ). DSP after splitting By reducing the dimensionality of the ultra-large-scale linear programming problem into multiple independent smaller problems, the solution efficiency is further improved. DSP It is expressed as follows: Objective function:

[0068] Constraints:

[0069] If SP is not feasible, DSP is unbounded, which means it is fixed. This is an invalid solution to the original problem. At this point, the DSP will return a polar ray. ,Right now , , , , , , , and , , It is a set of polar rays. Each ray represents a direction in the dual feasible region that allows the objective function to grow infinitely, and a feasible cut can be constructed using Farkas' lemma. The feasible cut is as follows:

[0070] In optimization theory, the objective function value of any feasible solution is an upper bound of the optimal objective function value. Therefore, if SP is feasible and bounded, According to the theory of strong duality, we have = .So, At this point, the dual variable vector returned by the DSP is a pole. p ,Right now , , , , , , , , This pole contains the gradient information of the subproblem's value function, which can be used to construct the optimal cut. The optimal cut is as follows:

[0071] in, P This represents the set of poles in the feasible region of the DSP. . Representing a scene The estimated cost of a subproblem is an approximation of the expected cost of the subproblem, and also serves as a carrier for feedback information from the subproblem to the main problem. . By continuously accepting optimal cut constraints from subproblems, we gradually approach the lower bound of the true value function of SP. The role of the optimal cut is to continuously raise the value of SP. The lower bound forces the main problem to search for integer variables that can truly reduce the expected cost of the subproblems. .

[0072] The main problem focuses on optimizing the integer part, with the goal of minimizing the total cost. Based on the splitting rules, the initial main problem is as follows: Objective function:

[0073] Constraints:

[0074] This is a lower bound estimate of the cost of the subproblems, initially completely open. Therefore, the initial primal problem can only provide an initial trial solution and an optimistic lower bound. , Therefore, it is necessary to continuously search for feasible regions or raise the lower bound based on dual information. Therefore, when the DSP is unbounded, a feasible cut is added to the main problem, removing those that make the subproblems infeasible. When the DSP reaches the optimum, an optimal cut is added to the main problem to tighten the solution. The lower bound ensures that it no longer underestimates the cost of subproblems, thus yielding new... Through repeated iterations, Gradually approaching the true cost, until The complete main problem (MP) is as follows: Objective function:

[0075] Constraints:

[0076] (3.2) Column generation In the Benders decomposition algorithm described above, MP needs to simultaneously optimize the core hub selection variable ( ), Backup hub selection variables ( ), path selection variables ( ) and backup hub activation variables ( Since these variables are all integers, and the number of their combinations grows exponentially with the size of the network nodes, directly enumerating all possible network configurations in MP leads to a severe curse of dimensionality, resulting in extremely low solution efficiency. To overcome this bottleneck, this invention introduces the Dantzig-Wolfe decomposition concept, reconstructing MP into a network configuration selection problem.

[0077] Restricted Master Problem (RMP) Based on Column Generation definition This is a structurally feasible network configuration scheme. Each scheme... A set of decision variable vectors satisfying constraints (10)-(21), (23)-(25), and (27) is denoted as It is worth noting that, although While valid in MP, this approach may fail to meet terminal demands due to capacity or storage limits in certain extreme random scenarios, leading to an unbounded dual problem and rendering SP infeasible.

[0078] Since it is impossible to enumerate all feasible solutions at once, a column pool is introduced. It only includes currently generated valid solutions, and initially contains only a small number of valid solutions. In other words, when a certain solution... When the test statistic calculated for the pricing subproblem is negative, the solution is considered efficient and can be included in the calculation. middle.

[0079] set up This indicates whether to select a plan. The Restricted Master Problem (RMP) based on column generation aims to improve column pooling. The proposed solution minimizes the expected total cost of the system while satisfying both Benders optimal cut and feasible cut.

[0080] The initial mathematical model for RMP is as follows.

[0081] Objective function:

[0082] Constraints:

[0083] If the DSP is unbounded, the inner loop of column generation will not be entered, and a feasible cut will be added to the RMP:

[0084] This constraint is used to remove combinations of solutions that make subproblems infeasible. For the set of polar rays that lead to the unbounded duality problem. Coefficients Using polar rays The plan was quantified. The degree of infeasibility. The formula for calculation is:

[0085] If the DSP finds the optimal solution, it enters the column generation inner loop and adds the optimal cut to the RMP:

[0086] This constraint is used to approximate the cost of SP. Wherein, the coefficients... Quantify the selection options At that time, the scheme at a specific dual pole The contribution value of the left-hand side of the optimal cut is calculated using the following formula:

[0087] Solving the RMP with the optimal cut allows us to obtain the dual variables. , These are the dual variables of constraints (66) and (70), respectively. Then... Substitute this into the pricing sub-problem (PP) to find new columns (i.e., new solutions) that can improve the current expected cost. ).

[0088] Pricing Sub-problem (PP) PP is a mixed-integer programming problem with the objective of minimizing the number of tests. Its decision variables are the variables required to generate new solutions. The model is as follows:

[0089] Column generation inner loop solution steps ①Step 1: Solve the relaxation RMP. In the current column pool Solve the linearly restricted principal problem to obtain the optimal solution. and dual variables .

[0090] ②Step 2: Solve for PP. The dual variable... Given a pricing problem, find the minimum test number. and its corresponding optimal solution .

[0091] ③Step 3: Convergence judgment and addition. If This indicates that a better solution exists. Add to column pool Return to Step 1. If This indicates that the current relaxed RMP has reached its optimal level, and the cost cannot be reduced by adding a new scheme. The column generation process terminates.

[0092] ④Step 4: Obtaining Integer Solutions. After the column generation converges, restore the integer constraints in RMP to obtain the integer optimal solution to the constrained master problem. This feedback is then sent to the outer Benders loop for feasibility and optimality verification.

[0093] In summary, the Benders-CG solution process can be found here. Figure 4 As shown in Table 1, the solution steps are as follows.

[0094] Table 1. Steps for solving the Benders-CG algorithm

[0095] In summary, the Benders-CG hybrid acceleration algorithm, which combines Benders decomposition and column generation, rapidly obtains the global optimal solution to large-scale stochastic mixed integer programming problems through collaborative solving of inner and outer double-layer loops.

[0096] This invention can be a system, method, and / or computer program product. A computer program product may include a computer-readable storage medium having computer-readable program instructions loaded thereon for causing a processor to implement various aspects of the invention.

[0097] For example, a joint optimization method for multimodal transport networks and inventory considering uncertainty constructs an expected comprehensive cost function and constraints based on the multimodal transport network and inventory. The expected comprehensive cost includes transportation costs, warehousing costs, stockout costs, and the activation cost of temporarily activating a backup hub as a core hub. The constraints include constraints based on a dynamic hub replacement mechanism, which involves pre-setting backup hubs around the core hub, and these backup hubs are capable of warehousing and transshipment conversion. Under these constraints, Benders decomposition and column generation are used to solve for decision variables that minimize the expected comprehensive cost. These decision variables include: a core hub selection variable. : Select a node j As a core hub, , For the set of potential core hubs; alternative hub selection variables : Select a node i As a core hub j backup hub , W For the set of potential backup hubs; path selection variables Within period t, the scenario Should a transportation method be used? From node To the node transportation, , , , For a set of nodes, For the collection of transportation modes, For a periodic set, A collection of scenarios; backup hub activation variable In the scene Next, cycle Internal backup hub Whether it will be activated as a temporary core hub ; Flow allocation variables In the cycle Inside, scene Next node To the node By transportation Freight volume of transportation , , , Inventory quantity variable Scene Next, node In the cycle Ending inventory , DFor the set of demand points; for the stockout quantity variable Scene Next, node In the cycle Stockouts at the end of the period, .

[0098] For example, a multimodal transport network and inventory joint optimization system considering uncertainty includes: a model building module configured to construct an expected comprehensive cost function and constraints based on the multimodal transport network and inventory, wherein the expected comprehensive cost includes transportation costs, warehousing costs, stockout costs, and activation costs of temporarily activating a backup hub as a core hub; the constraints include constraints designed based on a dynamic hub replacement mechanism, wherein the dynamic hub replacement mechanism involves pre-setting backup hubs around the core hub, and the backup hubs are capable of warehousing and transshipment conversion; and a solution module configured to solve for decision variables that minimize the expected comprehensive cost under the constraints, wherein the decision variables include: a core hub selection variable. : Select a node j As a core hub, , For the set of potential core hubs; alternative hub selection variables : Select a node i As a core hub j backup hub , W For the set of potential backup hubs; path selection variables Within period t, the scenario Should a transportation method be used? From node To the node transportation, , , , For a set of nodes, For the collection of transportation modes, For a periodic set, A collection of scenarios; backup hub activation variable In the scene Next, cycle Internal backup hub Whether it will be activated as a temporary core hub ; Flow allocation variables In the cycle Inside, scene Next node To the node By transportation Freight volume of transportation , , , Inventory quantity variable Scene Next, node In the cycle Ending inventory , D For the set of demand points; for the stockout quantity variable Scene Next, node In the cycle Stockouts at the end of the period, .

[0099] Computer-readable storage media can be tangible devices capable of holding and storing instructions for use by an instruction execution device. Computer-readable storage media can be, for example, but not limited to, electrical storage devices, magnetic storage devices, optical storage devices, electromagnetic storage devices, semiconductor storage devices, or any suitable combination thereof. More specific examples (a non-exhaustive list) of computer-readable storage media include: portable computer disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), static random access memory (SRAM), portable compact disc read-only memory (CD-ROM), digital multifunction disc (DVD), memory sticks, floppy disks, mechanical encoding devices, such as punch cards or recessed protrusions storing instructions thereon, and any suitable combination thereof. The computer-readable storage media used herein are not to be construed as transient signals themselves, such as radio waves or other freely propagating electromagnetic waves, electromagnetic waves propagating through waveguides or other transmission media (e.g., light pulses through fiber optic cables), or electrical signals transmitted through wires.

[0100] The computer-readable program instructions described herein can be downloaded from computer-readable storage media to various computing / processing devices, or downloaded via a network, such as the Internet, local area network, wide area network, and / or wireless network, to an external computer or external storage device. The network may include copper transmission cables, fiber optic transmission, wireless transmission, routers, firewalls, switches, gateway computers, and / or edge servers. A network adapter card or network interface in each computing / processing device receives the computer-readable program instructions from the network and forwards them to the computer-readable storage media in the respective computing / processing device.

[0101] The computer program instructions used to perform the operations of this invention may be assembly instructions, instruction set architecture (ISA) instructions, machine instructions, machine-dependent instructions, microcode, firmware instructions, state setting data, or source code or object code written in any combination of one or more programming languages, including object-oriented programming languages ​​such as Smalltalk, C++, Python, etc., and conventional procedural programming languages ​​such as "C" or similar languages. The computer-readable program instructions may be executed entirely on the user's computer, partially on the user's computer, as a standalone software package, partially on the user's computer and partially on a remote computer, or entirely on a remote computer or server. In cases involving a remote computer, the remote computer may be connected to the user's computer via any type of network—including a local area network (LAN) or a wide area network (WAN)—or may be connected to an external computer (e.g., via the Internet using an Internet service provider). In some embodiments, electronic circuitry, such as programmable logic circuitry, field-programmable gate arrays (FPGAs), or programmable logic arrays (PLAs), is personalized by utilizing state information from the computer-readable program instructions. This electronic circuitry can execute the computer-readable program instructions to implement various aspects of the invention.

[0102] The various embodiments of the present invention have been described above. These descriptions are exemplary and not exhaustive, nor are they limited to the disclosed embodiments. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the described embodiments. The terminology used herein is chosen to best explain the principles, practical application, or technical improvements to the embodiments in the market, or to enable others skilled in the art to understand the embodiments disclosed herein. The scope of the invention is defined by the appended claims.

Claims

1. A method for joint optimization of multimodal transport networks and inventory considering uncertainties, characterized in that: Based on the multimodal transport network and inventory, an expected comprehensive cost function and constraints are constructed. The expected comprehensive cost includes transportation cost, warehousing cost, stockout cost, and activation cost of the core hub when a backup hub is temporarily activated. The constraints include constraints designed based on a dynamic hub replacement mechanism, which involves pre-setting backup hubs around the core hub. These backup hubs are capable of warehousing and transshipment conversion. Under the constraints, the decision variables that minimize the expected total cost are solved using Benders decomposition and column generation. These decision variables include: Core Hub Selection Variables : Select a node j As a core hub, , A collection of potential core hubs; Backup hub selection variables : Select a node i As a core hub j backup hub , W A set of potential backup hubs; Path selection variables Within period t, the scenario Should a transportation method be used? From node To the node transportation, , , , For a set of nodes, For the collection of transportation modes, For a periodic set, A collection of scenes; Backup hub activation variable In the scene Next, cycle Internal backup hub Whether it will be activated as a temporary core hub ; Flow allocation variables In the cycle Inside, scene Next node To the node By transportation Freight volume of transportation , , , ; Inventory quantity variable Scene Next, node In the cycle Ending inventory , D For the set of demand points; Stockout quantity variable Scene Next, node In the cycle Stockouts at the end of the period, .

2. The method according to claim 1, characterized in that, The transportation methods include road transportation, rail transportation and water transportation.

3. The method according to claim 1, characterized in that, Under the constraints, the decision variables that minimize the expected total cost are solved using Benders decomposition and column generation. The steps include: The original problem is to minimize the expected comprehensive cost under the constraints. The Benders decomposition strategy is used to break down the original problem into a main problem and sub-problems. The decision variables of the main problem are core hub selection variables, backup hub selection variables, path selection variables, and backup hub activation variables. The variables of the sub-problems are flow allocation variables, inventory variables, and stockout variables. The main problem is converted into a constrained main problem based on columns, and the sub-problems are transformed into dual problems. Obtain a set of initial values ​​for the core hub selection variable, alternate hub selection variable, path selection variable, and alternate hub activation variable that satisfy the main problem. ; Step P1: Solve the main problem based on the initial solution, and obtain a fixed integer solution after a preset number of iterations. Using fixed integer solutions Solve the duality problem; If the solution to the dual subproblem is unbounded, then add a feasible cut to the restricted main problem; Step P2: Solve the main problem with integer constraints to obtain... Update the Nether ;use Update the initial solution and return to step P1; If the solution to the dual problem has an optimal solution, then the extreme point is obtained. p and target value Calculate the true total cost of the current solution, update the upper bound UB, and calculate the relative gap. ; If Gap is less than the preset convergence tolerance Output the optimal solution and end the algorithm; If Gap is greater than or equal to the preset convergence tolerance Add the optimal cut to the constrained principal problem; Solve the relaxed constraint master problem to obtain the dual variables; Based on the obtained dual variables, solve the pricing component problem and obtain the minimum test number. and its corresponding optimal solution ; like ,Will Add to current column pool ; like The column generation process ends; restore the integer constraints in the main problem and return to step P2.

4. The method according to claim 1, characterized in that, Constraints designed based on the hub dynamic substitution mechanism include: Among them: constraint (3) and constraint (4) are the flow conservation constraints of the core hub and the backup hub, respectively, to ensure that the inflow plus the beginning inventory equals the outflow plus the ending inventory. Constraint (6) Ensure that the holdings of core hubs, backup hubs, and demand points do not exceed their maximum physical storage capacity; Constraint (7) stipulates that when the standby hub is not activated, it acts as a warehouse and must maintain a certain minimum inventory level; Constraint (8) ensures that only standby hubs assigned to core hubs are allowed to hold inventory; Constraint (15) mandates that each selected core hub must be equipped with a backup hub; Constraint (16) stipulates that each backup hub can serve at most one core hub; Constraints (17) and (18) ensure that the transport path between the two exists only when the attribution relationship is established or the backup hub is activated; Constraint (21) ensures that backup hubs can only be assigned to core hubs that have already been selected; Constraints (23) and (24) work together with constraints (17) and (18) to allow direct transport with nodes other than the core hub only when the backup hub is activated; Constraint (25) ensures that only standby hubs that have already been assigned affiliations are eligible to be activated; Safety stock for backup hubs ; It is a pre-defined, extremely large number; Let W be the size of the set.

5. The method according to claim 4, characterized in that, Linearizing constraint (7) yields a simplified constraint: .

6. The method according to claim 3, characterized in that, Based on the Dantzig-Wolfe decomposition idea, the main problem is reconstructed into a network configuration scheme selection problem, and then the main problem is further constrained by generating a column-based main problem.

7. A computer-readable storage medium, characterized in that: The computer program is stored that can be loaded by a processor and executed according to any one of claims 1 to 6.

8. A multimodal transport network and inventory joint optimization system considering uncertainty, characterized in that, include: The model building module is configured to construct an expected comprehensive cost function and constraints based on the multimodal transport network and inventory. The expected comprehensive cost includes transportation costs, warehousing costs, stockout costs, and activation costs of temporarily activating backup hubs as core hubs. The constraints include constraints designed based on a dynamic hub replacement mechanism, which involves pre-setting backup hubs around the core hub. These backup hubs are capable of warehousing and transshipment conversion. The solution module, configured under the constraints, solves for the decision variables that minimize the expected total cost, the decision variables including: Core Hub Selection Variables : Select a node j As a core hub, , H A collection of potential core hubs; Backup hub selection variables : Select a node i As a core hub j backup hub , W A set of potential backup hubs; Path selection variables Within period t, the scenario Should a transportation method be used? From node To the node transportation, , , , For a set of nodes, For the collection of transportation modes, For a periodic set, A collection of scenes; Backup hub activation variable In the scene Next, cycle Internal backup hub Whether it will be activated as a temporary core hub ; Flow allocation variables Within period t, the scenario Next node To the node By transportation m Freight volume of transportation , , , ; Inventory quantity variable Scene Next, node In the cycle Ending inventory , D For the set of demand points; Stockout quantity variable Scene Next, node In the cycle Stockouts at the end of the period, .

Images